Hamiltonicity of Maximal Planar Graphs and Planar Triangulations

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Hamiltonicity of maximal planar graphs and planar triangulations

Von der Fakult¨at fur¨ Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westf¨alischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von

Diplom-Mathematiker Guido Helden

aus Aachen

Berichter: Professor Dr. Yubao Guo Universit¨atsprofessor Dr. Dr. h.c. Hubertus Th. Jongen

Tag der mundlic¨ hen Prufung:¨ 26.06.2007

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨

Preface

One of the typical topics in graph theory is the study of planar graphs. Planar graphs arise quite naturally in real-world applications, such as road or railway maps and chemical molecules. The cartographers of the past were aware of the fact that any map on the plane could be colored with four or fewer colors so that no adjacent countries were colored alike. It was the Four Color Problem which stimulated interest in hamiltonian planar graphs. A graph is said to be hamiltonian if it has a cycle that contains all vertices exactly once. If a planar graph is hamiltonian, then it is easy to color its faces with four or fewer colors so that no two adjacent faces are colored alike. Moreover, planar graphs play an important role as some practical problems can be eﬃciently solved for planar graphs even if they are intractable for general graphs.

The study of planar graphs was initiated by L. Euler in the 18th century. It was sub- sequently neglected for 180 years and only came to life again with Kuratowski’s theorem, which provides a criterion for a graph to be planar. At the same time, one of the most outstanding results for maximal planar graphs was proved by H. Whitney. A maximal planar graph is a planar graph having the property that no additional edges can be added so that the resulting graph is still planar. In 1931 H. Whitney [55] proved that each maximal planar graph without separating triangles is hamiltonian; a separating triangle is a triangle whose removal separates the graph. In particular, all 4–connected maximal planar graphs are hamiltonian.

Another connection of maximal planar graphs and the Four Color Problem is the following. In 1884 P.G. Tait [49] conjectured that every cubic, 3–connected and planar graph is hamiltonian. Note that the dual graph of a cubic, 3–connected and planar graph is a maximal planar graph. The Conjecture of Tait became more important, in the sense that its correctness would imply the positive correctness of the Four Color Problem. Un- fortunately, the latter conjecture is wrong, as shown by W.T. Tutte in 1946. Therefore, the Four Color Problem is true if and only if every hamiltonian planar graph is 4–colorable.

In 1956 W.T. Tutte [53] generalized Whitney’s result and proved that each 4–connected planar graph is hamiltonian. In 1990 M.B. Dillencourt [24] generalized Whitney’s result in a diﬀerent direction, namely to graphs called NST–triangulations. M.B. Dillencourt

i ii deﬁned NST–triangulations to be those planar graphs that are triangulations in the sense that every face, perhaps except for the exterior face, is a triangle but that there is no separating triangle. In the class of the 3–connected maximal planar graphs numerous ex- amples which are non-hamiltonian were found. In 2003 C. Chen [17] obtained a stronger version of Whitney’s result and he proved that any maximal planar graph with only one separating triangle is hamiltonian. An interesting problem with respect to the class of the 3–connected maximal planar graphs is the following: What is the maximal number α, so that every maximal planar graph with at most α separating triangles is hamiltonian?

In this thesis we will extend the results of C. Chen and M.B. Dillencourt in various ways. We will mainly study the existence of hamiltonian cycles and hamiltonian paths in maximal planar graphs and planar triangulations. We will prove that each maximal planar graph with at most ﬁve separating triangles is hamiltonian. In the case of more separating triangles, we will introduce a special structure of the position of the separating triangles to each other. The latter structure will also generate hamiltonicity.

Chapter 1 will be devoted to an introduction of the terminology and the basic concepts of planar graphs and maximal planar graphs. We will introduce one of the most powerful tools in the theory of planar graphs, namely Euler’s formula. This concept will be the corner-stone of many results of us.

In Chapter 2 we will study some fundamentals of maximal planar graphs. The ﬁrst section will be concerned with the generation of some classes of maximal planar graphs. There are two main reasons why this generation will be of interest: On the one hand, this will provide a basis for inductive proof; and on the other hand, this can be used to develop eﬃcient algorithms for the constructive enumeration of the structures. In the second section we will focus on the structures of some classes of maximal planar graphs. Moreover, we will study the subgraphs of a maximal planar graph G induced by vertices of G with the same degree. In the last section we will turn towards the connectivity of maximal planar graphs.

The most interesting problem regarding hamiltonian maximal planar graphs will be investigated in Chapter 3. In the first section we will extend the result of Chen [17]. We will prove that each maximal planar graph with at most five separating triangles is hamiltonian. We will give a counterexample for the case of six separating triangles. In the case of more separating triangles, we will establish a special structure of the position of the separating triangles to each other, which will also generate hamiltonicity. In the second section we will deal with hamiltonian paths. Since maximal planar graphs with more than five separating triangles need not be hamiltonian, we will show that maximal planar graphs with exactly six separating triangles have at least one hamiltonian path. It would be nice to have a theorem that says: “A graph G is non-hamiltonian if G has iii property ‘Q’, where ‘Q’ can be checked in polynomial time”. This motivates the concept of 1–toughness, which is introduced in the fourth section. In the fifth section we will construct specific maximal planar graphs. We will show that for all n ≥ 13 and k ≥ 6, there exist a hamiltonian as well as a non-hamiltonian maximal planar graph with n vertices and k separating triangles. In the sixth section we will study the number of cycles of certain length that a maximal planar graph G on n vertices may have. In the seventh section we will consider pancyclicity rather than hamiltonicity. In fact, we will determine whether a maximal planar graph G contains a cycle of each possible length l with 3 ≤ l ≤ n. In the eighth section we will deal with the question: How many vertices of a hamiltonian maximal planar graph can be deleted, so that the remaining graph is still hamiltonian?

In Chapter 4 we will turn to general planar triangulations. Clearly, each maximal planar graph is a planar triangulation. In the ﬁrst section we will consider some properties of planar triangulations. The second section of this chapter deals with general results on 2–connected planar triangulation. In the third section we will see that the deletion of vertices of a hamiltonian maximal planar graph leads to some interesting results in the class of 3–connected planar triangulations. In the last section we will concentrate on 4–connected planar triangulation.

In Chapter 5 we will consider some applications of hamiltonian maximal planar graphs and planar triangulations. In the ﬁrst section of this chapter we will turn towards algorithms, which will be useful for applications. In the second section we will show that the dual graph of a maximal planar graph is of interest for applications. In the third section we will deal with some applications of hamiltonian maximal planar graphs and planar triangulations in computer graphics. In the last section we will focus on applications in chemistry.

I am particularly indebted to my advisor, Prof. Dr. Y. Guo, for introducing me to graph theory and for giving me the opportunity to conduct my doctorial studies under his supervision. I also would like to express my deep gratitude to Prof. Dr. Dr. h.c. H. Th. Jongen for acting as co-referee and for his support during the last few years. Many thanks also to my colleague Jinfeng Feng for stimulating discussions and excellent teamwork.

Finally, I would like to express my appreciation for my wife and two children, who allowed me take the time to write this thesis.

Contents

1 Introduction 1 1.1 Terminology and notation ...... 1 1.2 Subject ...... 3 2 Fundamentals of maximal planar graphs 5 2.1 Generating maximal planar graphs ...... 5 2.2 Properties of maximal planar graphs ...... 8 2.2.1 Properties of MPG-3 graphs ...... 10 2.2.2 Properties of MPG-4 graphs ...... 12 2.2.3 Properties of MPG-5 graphs ...... 16 2.3 Connectivity of maximal planar graphs ...... 17 3 Hamiltonicity of maximal planar graphs 21 3.1 Hamiltonian cycle ...... 21 3.2 Hamiltonian path ...... 39 3.3 Length of a hamiltonian walk ...... 40 3.4 Tough graphs ...... 41 3.5 Construction of maximal planar graphs ...... 42 3.6 Number of cycles of certain length ...... 50 3.6.1 Number of short cycles ...... 50 3.6.2 Number of longer cycles ...... 51 3.6.3 Number of hamiltonian cycles ...... 51 3.7 Pancyclic maximal planar graph ...... 55 3.8 Deletion of vertices ...... 57 3.9 Claw-free maximal planar graph ...... 65 4 Hamiltonicity of planar triangulations 69 4.1 Properties of planar triangulations ...... 69 4.2 2–connected planar triangulations ...... 71 4.3 3–connected planar triangulations ...... 73 4.4 4–connected planar triangulations ...... 77 5 Applications of PT and MPG 79 5.1 Algorithms ...... 79 5.2 Dual graphs of maximal planar graphs ...... 85 5.3 Applications in computer graphics ...... 87 5.3.1 Interior dual hamiltonian planar triangulations of polygons . . . . . 90 5.3.2 Interior dual hamiltonian planar triangulations of point sets . . . . 92 5.3.3 Sequential planar triangulation ...... 92 5.3.4 Graham triangulation ...... 93 5.3.5 Hamiltonian cycles in 3–dimensional polygons ...... 94 5.4 Applications in Chemistry ...... 95 Bibliography 97

Chapter 1

Introduction

In this chapter we present the basic notation and terminology of graph theory which will be used throughout this thesis. Before discussing results related to maximal planar graphs in more detail in the second section, we will briefly explain the basic definitions and concepts in the first section. The notation mainly follows that of L. Volkmann [54] as well as G. Chartrand and L. Lesniak [16] and we refer the reader to these books for any information not provided here. Special notation and definitions will be defined where needed.

1.1 Terminology and notation

General concepts

If not stated otherwise, the term graph will be used throughout the thesis to represent a ﬁnite, simple and undirected graph. We denote the vertex set of a graph G with V (G) and the edge set with E(G). The cardinalities of these sets will be the order n(G) and the size m(G) of G, respectively. If two vertices u, v ∈ V (G) are connected with an edge, we simply write uv ∈ E(G) and say that u and v are adjacent. An edge e = uv is called incident with both end-vertices u and v. Two distinct edges with a common end-vertex are called adjacent edges.

A cycle with the vertices x1, x2, ..., xk and the edges x1x2, x2x3, ..., xkx1 is called a k- cycle and is denoted by x1x2...xkx1. A path from x to y is called a P (x, y) path.

The neighborhood NG(v) of a vertex v is deﬁned as NG(v) = {u ∈ V (G) | uv ∈ E(G)}. The degree dG(v) of a vertex v is deﬁned as the number of edges incident with v. If dG(v) = 0, we call v an isolated vertex. The minimum degree δ(G) and the maximum degree ∆(G) of a graph G are the minimum and maximum over all vertex-degrees in G, respectively. τi(G) of a graph G denotes the number of all vertices with degree i. If δ(G) = ∆(G) = d, we call the graph G d–regular. A 3–regular graph G is also called cubic.

A graph H is a subgraph of a graph G if V (H) ⊆ V (G) and E(H) ⊆ E(G). The subgraph H is called induced if E(H) = {uv ∈ E(G) | u, v ∈ V (H)}. For a vertex subset

1 2 CHAPTER 1. INTRODUCTION

S ⊆ V (G), G [S] is the subgraph of G induced by S. The deletion of a vertex x in G, in symbols G − x, denotes the induced subgraph G [V (G) \ {x}]. The deletion of a set of vertices S is deﬁned as G − S = G [V (G) \ S]. Let u and v be two vertices of G. If uv ∈ E(G), then G + uv = G; otherwise G + uv denotes the graph obtained from G by adding the edge uv.

A graph G is homomorphic to a graph H if a mapping g : V (G) → V (H) exists, which is called homomorphism, so that g preserves adjacency. That is, xy ∈ E(G) if and only if g(x)g(y) ∈ E(H). If g is a one-to-one mapping, g is called isomorphism and we call the two graphs G and H isomorphic, in symbols G ∼= H.

Connectivity

A vertex x is said to be connected to a vertex y in a graph G if there exists a path between these two vertices in G. A graph G is called connected if it contains a path between any two vertices. All vertices which are connected to each other induce a component, and κ(G) denotes the number of components of G. Note that if κ(G) = 1, then G is connected. A k–cut of a graph G is a set S of k vertices of a graph G so that G − S is not connected. Note that a k–cut is also called a separating set of k vertices. The connectivity σ(G) of a graph G is the size of a smallest k–cut of G if G is not complete, and σ(G) = n − 1 if G = Kn for any positive integer n. Hence σ(G) is the minimum number of vertices whose removal results in a disconnected or a trivial graph. A graph G is said to be k–connected, k ≥ 1 if σ(G) ≥ k. In general, G is k–connected if and only if the removal of fewer than k vertices results in neither a disconnected nor a trivial graph.

Planar graphs

A graph G is said to be embeddable on a surface S if it can be drawn on S in a way that no two edges intersect geometrically except at a vertex, at which they are both incident. In this thesis we are concerned exclusively with the case in which S is a plane or sphere. A graph G is called planar if it can be embedded in the plane. Note embedding a graph in the plane is equivalent to embedding it on the sphere. A planar graph divides the plane into regions, which are called faces. The cardinality of the set of all faces is denoted by l(G). The unbounded region is called exterior face, the other faces are called interior faces. In a 2–connected planar graph G, the exterior cycle is the cycle bounding the exterior face and will be denoted by XG. Cycles bounding an interior face are called facial cycles. An edge is called a boundary edge of a graph G if it is on the exterior cycle; otherwise it is an interior edge. A vertex is called a boundary vertex of a graph G if it is on the exterior cycle. A vertex that is not a boundary vertex is called an interior vertex. A chord is an interior edge whose end vertices are both boundary vertices.

A planar graph G is called maximal planar if for every pair u, v of non-adjacent vertices of G the graph G + uv is non-planar. Thus in any embedding of a maximal planar graph G with |V (G)| ≥ 3 vertices, the boundary of every face is a triangle. 1.2. SUBJECT 3

A planar graph is called planar triangulation if the boundary of every face is a triangle, except possibly the exterior face. A graph G is called triangulated if every cycle of G with length greater than three has a chord.

A separating k–cycle is a k–cycle so that both the interior as well as the exterior contain one or more vertices. A separating 3–cycle is called a separating triangle. Therefore, a separating triangle does not form the boundary of a face.

For a given connected planar graph G we deﬁne the dual graph G∗ as follows: All faces of G constitute the vertex set of G∗. Two distinct vertices of G∗ are joined by an edge if the corresponding faces in G have a common edge as a border. Each edge of G∗ is drawn so that it crosses its associated edge of G but no other edge of G or G∗.

Hamiltonicity

A cycle of a graph G is called a hamiltonian cycle if it contains all vertices of G. A graph G is deﬁned to be hamiltonian if it has a hamiltonian cycle. Similarly a hamiltonian path is a path of a graph G which contains all vertices of G.

An embedding of a maximal planar graph G is called hamiltonian for any two boundary edges of G if for any two boundary edges of G, the graph G has a hamiltonian cycle passing through these two edges. A maximal planar graph G is called hamiltonian for any two boundary edges of G if for each embedding of G, the graph G is hamiltonian for any two boundary edges of G.

1.2 Subject

One of the main topics in graph theory is the concept of planarity. In the 18th century, Euler made one of the ﬁrst discoveries in this ﬁeld, the well known “Polyederformel”.

Theorem 1.1 (Euler [27]) Let G be a planar graph on n vertices and m edges. Then

l(G) = 1 + µ(G) = 1 + m(G) − n(G) + κ(G).

Since planar triangulations are always connected, we can use the following observation.

Observation 1.2 Let G be a connected planar graph on n vertices and m edges. Then

l(G) = 2 + m(G) − n(G).

According to Theorem 1.1 of Euler, each embedding of a planar graph has the same number of faces.

It was Kuratowski [38] who gave a characterization of planar graphs in terms of for- bidden homeomorphic subgraphs. We say a graph H is homeomorphic from G if either H is isomorphic to G or H is isomorphic to a graph obtained by subdividing any sequence 4 CHAPTER 1. INTRODUCTION of edges of G. Note that by subdivision of an edge e = xy of a non-empty graph G, we mean that the edge xy is removed from the graph and a new vertex w is inserted instead, along with the edges wx and wy. We say G is homeomorphic with H if both G and H are homeomorphic from a graph F . Theorem 1.3 (Kuratowski [38]) A graph is planar if and only if it possesses no subgraph homeomorphic with either K5 or K3,3. Now we will study some elementary properties of maximal planar graphs. Theorem 1.4 (Chartrand and Lesniak [16]) Let G be a maximal planar graph on n ≥ 3 vertices and m edges. Then 2 l(G) = m(G) and m(G) = 3n(G) − 6. 3 This result provides the next corollary for planar triangulations, in order to answer the following question: How many edges has a planar triangulation? Corollary 1.5 Let G be a planar triangulation on n ≥ 3 vertices and m edges. Then 2n(G) − 3 ≤ m(G) ≤ 3n(G) − 6. In the inequality of the left side the equality holds if every vertex of G is on the exterior face. Moreover, Theorem 1.4 yields the following observation. Observation 1.6 Let G be a maximal planar graph on n ≥ 3 vertices. Then l(G) = 2n(G) − 4. Thus we obtain a necessary condition for a planar graph to be maximal planar. If a planar graph G has an odd number of faces, then G is not maximal planar. Another important consequence of Theorem 1.1 of Euler is the following corollary. Corollary 1.7 Let G be a maximal planar graph on n ≥ 4 vertices. Then

3τ3 + 2τ4 + τ5 = (7 − 6)τ7 + (8 − 6)τ8 + ... + (∆(G) − 6)τ∆(G) + 12. The latter corollary leads to an immediate but important consequence. Corollary 1.8 Every planar graph G contains a vertex of degree at most 5, that is, δ(G) ≤ 5. An interesting feature of maximal planar graphs on n ≥ 4 vertices is the fact that the minimum degree is at least 3. Together with the previous corollary we obtain δ(G) ∈ {3, 4, 5} in a maximal planar graph G of order at least four. This leads to a very interesting observation. Observation 1.9 Let G be a planar graph. Then G is at most 5–connected. While considering the stereographic projection, Chiba and Nishizeki showed the following lemma, which will be used in some proofs in Chapter 3. Lemma 1.10 (Chiba and Nishizeki [18]) Let G be a planar graph. Then G can always be embedded in the plane so that a given face of G becomes the exterior face. Consider the stereographic projection and rotate the sphere so that the north pole is in that face. Chapter 2

Fundamentals of maximal planar graphs

2.1 Generating maximal planar graphs

In this section we deal with the generation of some classes of maximal planar graphs, which will provide a basis for inductive proofs.

An inductive class I is deﬁned by giving:

• initial speciﬁcations which deﬁne the class A of initial elements – the basis of I.

• generating speciﬁcations which deﬁne the class R of rules (modes) of combination, any such rule applied to an appropriate sequence of elements, already in A, produces an element of A.

The inductive class I = Cn(A; R) consists exactly of the elements which can be obtained (constructed) from the basis by a ﬁnite number of applications of the generating rules.

There are two main reasons why methods to construct an infinite class from a finite set of starting graphs are of interest. On the one hand, this provides a basis for inductive proof; and on the other hand, this can be used to develop efficient algorithms for the constructive enumeration of the structures.

In the following inductive deﬁnitions, the starting graphs and rules should be under- stood as embedded on a surface. The small black triangles attached to the vertices in the description of the rule denote any number (zero or more) of edges. The condition ≤ 2 means that there can be no more than two edges on that side. The condition 1, 2 means there should be one or two edges on that side, see Figure 2.2. In the following we shall use the half-edges to indicate that there must be an edge.

Deﬁnition 2.1 The set of all maximal planar graphs with minimum degree i is called MPG-i; and we denote the subset of MPG-i with n vertices by MPGn-i.

5 6 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

In 1967 Bowen and Fisk [12] presented an inductive deﬁnition of the class of all maximal planar graphs.

Theorem 2.2 (Bowen and Fisk [12]) The inductive class Cn(S; A), see Figure 2.1, is equal to the class of all maximal planar graphs.

Simplex (S)

A ≤ 2

Figure 2.1: Bowen and Fisk’s operations.

In 1989 Batagelj [8] implemented a similar inductive procedure to generate the class of all maximal planar graphs without vertices of degree three.

Theorem 2.3 (Batagelj [8]) The inductive class Cn(O; B, C), see Figure 2.2, is equal to the class of all maximal planar graphs without vertices of degree three.

We need the following deﬁnition, in order to get a better formulation of the following result.

Deﬁnition 2.4 A graph is called cyclically k-edge-connected if at least k edges must be deleted to disconnect a component so that every remaining component contains a cycle.

Barnette [9] and Butler [14] independently described a method for constructing all planar cyclically 5-edge-connected cubic graphs. In the language of the dual graph this class is the set of all 5–connected maximal planar graphs. We call such maximal planar graphs 5–connected MPG-5 graphs. 2.1. GENERATING MAXIMAL PLANAR GRAPHS 7

Octrahedron (O)

1, 2

Figure 2.2: Batagelj’s operations. 8 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

Note that a maximal planar graph with minimum degree ﬁve is always a 3–connected MPG-5 graph, a 4–connected MPG-5 graph if there are no separating triangles and a 5–connected MPG-5 graph if there are no separating triangles or separating 4–cycles.

Barnette and Butler’s method starts with the icosahedron graph and uses the operations called D, E and F given in Figure 2.3. Since both proved an inductive deﬁnition of planar cyclically 5-edge-connected cubic graphs, the operations depictured here are just the dual forms of the operations used by Barnette and Butler.

Theorem 2.5 (Barnette [9] and Butler [14]) The inductive class Cn(I; D, E, F), see Fig- ure 2.3, is equal to the class of all 5–connected maximal planar graphs with minimum degree ﬁve.

Batagelj [6] described a method for constructing all 3–connected MPG-5 graphs. He uses the operations D and E also used by Barnette and Butler, and in addition, a switching operation G. This operation assumes that the top and bottom vertices do not share an edge.

Theorem 2.6 (Batagelj [6]) The inductive class Cn(I; D, E, G), see Figure 2.4, is equal to the class of all 3–connected maximal planar graphs with minimum degree ﬁve.

We denote an operation G such that the central edge does not belong to a separating triangle after the operation by an operation G3. Brinkmann and McKay [13] described a method for constructing all 4–connected MPG-5 graphs.

Theorem 2.7 (Brinkmann and McKay [13]) The inductive class Cn(I; D, E, G3) is equal to the class of all 4–connected maximal planar graphs with minimum degree ﬁve.

An important subclass of all 5–connected MPG-5 graphs, with many practical applications, are those with maximum degree 6, best known via their dual graphs, the fullerene graphs, see Chapter 5.

2.2 Properties of maximal planar graphs

In this section we analyse the structure of some classes of maximal planar graphs. More- over, we study the subgraphs of G induced by vertices of G with the same degree.

Deﬁnition 2.8 Let G = (V, E) be a graph.

Di = {x ∈ V |d(x) = i} denotes the vertex set of G with degree i. 2.2. PROPERTIES OF MAXIMAL PLANAR GRAPHS 9

Icosahedron (I)

D E

Figure 2.3: Barnette and Buttler’s operations.

Figure 2.4: Switching operation. 10 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

2.2.1 Properties of MPG-3 graphs We start with the following observation.

Observation 2.9 If G = (V, E) is an arbitrary maximal planar graph with δ(G) = 3, then G has the K4 as a subgraph.

Proof. Each MPG-3 graph has a vertex e of degree 3. Let the vertices a, b and c be the neighbors of e, see Figure 2.5. Then a, b and c are pairwise adjacent, since G is maximal planar. If not, then e has a further neighbor, which yields a contradiction to d(e) = 3.

c c

e e

b a b a

Figure 2.5: Each MPG-3 graph has a subgraph isomorphic to K4.

This observation has an easy corollary.

Corollary 2.10 If G = (V, E) is a maximal planar graph with δ(G) = 3 and n ≥ 5, then G has at least one separating triangle.

Now we study the subgraph of G induced by vertices of G with degree three.

Theorem 2.11 Let G = (V, E) be a maximal planar graph with δ(G) = 3. All components of the subgraph G [D3] belong to one of the following kind of graphs:

• K4 if n = 4, • trivial graph if n ≥ 5.

Proof. Let G = (V, E) be a maximal planar graph with δ(G) = 3. Assume x and y are two adjacent vertices with degree 3. Let the vertices a, b be the neighbors of x, and let the vertices c, d be the neighbors of y, see Figure 2.6. If the vertices a and c are distinct, then either x or y is adjacent to another vertex, since all faces of G are triangles, then the degree of this vertex is greater than three. Therefore, the vertices a and c are equal. We call this vertex e. With the same argumentation as above we show that the vertices b and d are equal. We call this vertex f. 2.2. PROPERTIES OF MAXIMAL PLANAR GRAPHS 11

a c e

x y x y

b d f

Figure 2.6: The subgraph D3.

Case 1: Let n = 4. Since G is maximal planar, the vertices e and f are adjacent. Therefore, G [D3] = K4.

Case 2: Let n ≥ 5. Since G is maximal planar, there exists a vertex which is adjacent to x or y. This leads to a contradiction to the assumption. Thus all components of the subgraph G [D3] are isolated vertices.

This theorem and the following trivial observation lead to a very useful corollary.

Observation 2.12 If G = (V, E) is a maximal planar graph with δ(G) = 3 and n = 5, then τ3 = 2 and G has one separating triangle.

This is the only case in which there are more vertices of degree three than separating triangles. Therefore, the following result holds.

Corollary 2.13 If G = (V, E) is a maximal planar graph with δ(G) = 3 and n ≥ 6, then the neighbors of every vertex of degree 3 induce a separating triangle.

Remark. For this reason the number of vertices with degree three, τ3, is a lower bound of the number of separating triangles if n ≥ 6.

The number of vertices with degree three has still another important property. It supplies a necessary condition of whether a graph is hamiltonian.

Theorem 2.14 Let G = (V, E) be an arbitrary maximal planar graph with δ(G) = 3 and n ≥ 13. If 2τ3 > n, then G is non-hamiltonian.

Proof. Assume G is hamiltonian. Let C be a hamiltonian cycle of G and x be a vertex of degree 3. Let the vertices a and b be the neighbors of x on the hamiltonian cycle C. If we replace the edges ax and xb by the edge ab, then we obtain a hamiltonian cycle C 0 in the maximal planar graph G − x. If we delete all vertices of degree 3 in this way, we 12 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

D x b a

Figure 2.7: 2τ3 ≥ n. obtain a maximal planar graph G − D3. For two distinct vertices of degree 3 one obtains diﬀerent edges. After construction the hamiltonian cycle of G − D3 has at least τ3 edges. G − D3 consists of n − τ3 vertices. Therefore, τ3 ≤ n − τ3, i.e. 2τ3 ≤ n. Thus we get a contradiction to the hypothesis.

2.2.2 Properties of MPG-4 graphs While considering MPG-4 graphs we must distinguish two diﬀerent classes of MPG-4 graphs, those which are 3–connected and those which are 4–connected, see Section 2.3. Inspired by Observation 2.9, we now present a degree condition for maximal planar graphs with δ(G) = 4. Theorem 2.15 (Helden [35]) If G = (V, E) is an arbitrary maximal planar graph with δ(G) = 4, then there is no vertex of degree four with dG[D4](x) = 3.

Proof. Assume that there is a vertex a with degree four and dG[D4](a) = 3. This vertex is adjacent to three vertices with degree four, we call them b, c and d, as well as to one vertex with degree ≥ 5. We call this vertex z. Since G is maximal planar, all faces of the graph G are triangles. Because of that, the edge which is incident to z and a belongs to two triangles, to which the vertices a and z belong as well. The vertex a has degree four, thus it can be connected with no further edge. From this it follows that the vertex z is adjacent to b and d. The same argument holds for the edges ad and ab. This means the vertex d is adjacent to the vertex c and the vertex b is adjacent to the vertex c. Moreover, a vertex y with dG(y) ≥ 4 exists and y is adjacent to the vertex d. The vertex d can be connected with no further vertex, because dG(d) = 4. Thus y is adjacent to z, as well as y is adjacent to c. Since dG(c) = 4 and y is adjacent to the vertex c, it follows that y is adjacent to the vertex b, because the graph G is maximal planar. Since dG(z) ≥ 5, a vertex x exists with dG(x) ≥ 4. The vertex x is adjacent to the vertex b because the graph G is maximal planar. Thus we get a contradiction to dG(b) = 4 and the proof is complete.

Studying the subgraph G [D4], we obtained the following result. 2.2. PROPERTIES OF MAXIMAL PLANAR GRAPHS 13

Theorem 2.16 (Helden [35]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. If there exists a vertex x in the subgraph G [D4] with dG[D4](x) = 4, then ∆(G) = 4 and n(G) = 6.

Proof. We prove this theorem by using Theorem 2.15 and using the attribute that the graph G is maximal planar. We build the graph around the vertex x. Since dG[D4](x) = 4, the vertex x is adjacent to four vertices with degree four. We call these four vertices a, b, c and d. As the graph G is maximal planar we obtain that the vertex a is adjacent to the vertices b and d. With the same argument we get that the vertex c is adjacent to the vertices d and b. Assume the vertex a is adjacent to the vertex c, then the vertex d cannot be adjacent to b. Thus we have dG[D4](a) = dG[D4](b) = dG[D4](c) = dG[D4](d) = 3. This is not possible because of Theorem 2.15. Therefore, all vertices must have degree four in the subgraph G [D4]. Thus there must be another vertex e with degree four. Now we have obtained all six vertices with degree four. An extension is not possible.

The graph constructed in the proof of Theorem 2.16 is denoted by MPG6-4, see Figure 2.8.

Figure 2.8: An MPG6-4 graph.

Next we will determine the structure of components of the subgraph G [D4].

Theorem 2.17 (Helden [35]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. All components of the subgraph G [D4] belong to the following types of graphs: • trivial graph,

• cycle Cp with p ≥ 5 with p ∈ IN, • path of length q with q ∈ IN,

• MPG6-4 graph.

Proof. The vertices in the subgraph G [D4] have maximal degree four. Because of Theo- rem 2.15 we know there is no vertex with degree three in the subgraph G [D4]. Therefore, the vertices in the subgraph G [D4] only have the degrees 0, 1, 2, or 4. If dG[D4](v) = 4 for 14 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

one vertex v, Theorem 2.16 supplies an MPG6-4 graph as a component and κ(G [D4]) = 1. If the vertices in the subgraph G [D4] have the degrees 0, 1 or 2 only the trivial graph, a path of length q with q ∈ IN or a cycle Cp with p ≥ 5 can occur as a component.

The next lemma provides properties which will be needed later.

Lemma 2.18 (Helden [35]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. If one component K of the subgraph G [D4] consists of a path of length q with q ∈ IN, then |NG(K)| = 4 and for all x ∈ NG(K) is d(x) ≥ 5.

Proof. A path of length q with q ∈ IN consists of q − 1 vertices with degree two in the subgraph G [D4] and two vertices with degree one in the subgraph G [D4]. The vertex x1 with dG[D4](x1) = 1 is adjacent to three vertices with degree at least ﬁve, we call this three vertices a, b and c, and x1 is adjacent to one vertex with degree four, we call this vertex x2. As the graph G is maximal planar, we get that the vertex c is adjacent to the vertices a and b and with the same argument we obtain that the vertex x2 is adjacent to the vertices a and b. If q ≥ 2 there exist q − 1 more vertices xi with degree four which are adjacent to xi−1, a and b, see Figure 2.9. The last vertex with degree four, we will call xq+1, satisﬁes dG[D4](xq+1) = 1. The vertex xq+1 is also adjacent to a and b. Now we have to prove that cxq+1 ∈/ E(G). Assume cxq+1 ∈ E(G).

Case 1: Let q ≤ 3. This yields a contradiction to dG(a) ≥ 5, dG(b) ≥ 5 or dG(c) ≥ 5.

Case 2: Let q ≥ 4. This yields a contradiction to c ∈/ G [D4]. This completes the proof of the lemma.

c x1 x2 xq xq+1

b Figure 2.9: Proof of Lemma 2.18

In analogy to Lemma 2.18 we can prove the next corollary.

Corollary 2.19 Let G = (V, E) be a maximal planar graph with δ(G) = 4. If one component K of the subgraph G [D4] consists of one cycle Cp with p ≥ 5 with p ∈ IN, then |NG(K)| = 2 and d(x) ≥ 5 for all x ∈ NG(K). 2.2. PROPERTIES OF MAXIMAL PLANAR GRAPHS 15

Since the neighborhood of a cycle component consists of only two vertices, adding one more vertex to the graph provides that the degree of at least one vertex of the cycle component increases to at least ﬁve. Hence, we get a path component. This means if one component of the subgraph G [D4] consists of one cycle, then no other component can exist. This yields the next easy corollary.

Corollary 2.20 (Helden [35]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. If one component of the subgraph G [D4] consists of one cycle Cp with p ≥ 5 with p ∈ IN, then κ(G [D4]) = 1.

The next lemma provides properties which will be needed later.

Lemma 2.21 (Helden [35]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. If the subgraph G [D4] consists of one cycle Cp with p ≥ 5 with p ∈ IN, then G contains exactly two vertices with degree at least ﬁve.

Proof. Let G [D4] consists of one cycle Cp with p ≥ 5 with p ∈ IN. We call the vertices of the cycle x1, x2, ..., xp. Since dG(xi) = 4 for all i = 1, 2, ..., p there exist at least two vertices a and b. Because the graph G is maximal planar, one vertex lies inside the cycle and one vertex lies outside the cycle. Furthermore both vertices a and b are adjacent to all vertices x1, x2, ..., xp. Because dG(xi) = 4 for all i = 1, 2, ..., p and the graph G is maximal planar, there exists no further vertex.

Now we consider the trivial components.

Corollary 2.22 (Helden [35]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. If one component K1 of the subgraph G [D4] consists of one vertex x, then |NG(x)| = 4 and d(y) ≥ 5 for all y ∈ NG(K1).

Now we consider 3–connected MPG-4 graphs. This means there exists at least one separating triangle. In 1978 Hakimi and Schmeichel showed the following result for the vertices with degree four in the interior of a separating triangle.

Theorem 2.23 (Hakimi and Schmeichel [30]) Let G = (V, E) be a maximal planar graph with δ(G) = 4. Suppose G contains a separating triangle induced by the set of vertices X = {x1, x2, x3}. Consider any planar embedding of G, and let Dini denote the vertices of degree i occurring in the interior of the triangle X in this embedding.

a) If |Din4| = 0, then |Din5| ≥ 7.

b) If |Din4| = 1, then |Din5| ≥ 5.

c) If |Din4| = 2, then |Din5| ≥ 3.

The octahedron, see Figure 2.2, yields the following observation. 16 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

Observation 2.24 Let G = (V, E) be a maximal planar graph with δ(G) = 4. Suppose G contains a separating triangle induced by the set of vertices X = {x1, x2, x3}. Consider any planar embedding of G, and let Dini denote the vertices of degree i occurring in the interior of the triangle X in this embedding.

If |Din4| = 3, then |Din5| ≥ 0.

Thus the previous results show that the number of vertices with degree 5 occurring in the interior of a separating triangle X depends on the number of vertices with degree 4 occurring in the interior of the same triangle X.

2.2.3 Properties of MPG-5 graphs In 1983 Batagelj showed that there is only one 5–regular maximal planar graph.

Theorem 2.25 (Batagelj [6]) Let G = (V, E) be a maximal planar graph with δ(G) = 5. If G = G [D5] with ∆(G) = 5, then this graph is unique up to isomorphism. We call this graph icosahedron or MPG12-5, see Figure 2.3.

Since maximal planar graphs with δ(G) = 5 and ∆(G) = 6 play a decisive role in their applications, we will show the next result. Rolland-Balzon [44] proved this result.

Theorem 2.26 (Rolland-Balzon [44]) Let G = (V, E) be a maximal planar graph with δ(G) = 5 and with 14 vertices. If ∆(G) = 6, then this graph is unique up to isomorphism. We call this graph MPG14-5, see Figure 2.10.

Theorem 2.27 Let G = (V, E) be a maximal planar graph with δ(G) = 5. Suppose G contains a separating 4–cycle induced by the set of vertices X = {x1, x2, x3, x4}. Then in any planar embedding of G, the interior and exterior of X must each contain at least seven vertices of degree at least ﬁve.

Proof. Let I(X) denote the set of vertices inside of the 4-cycle X. Let dI(X)(xi) denote the number of vertices in I(X) to which xi is adjacent, for i = 1, 2, 3, 4. Clearly dI(X)(xi) > 0 for each i, since G is maximal planar and I(X) =6 ∅. We can show that dI(X)(xi) ≥ 2 for every i, with dI(X)(xi) ≥ 3 for some i. That is

4 d (x ) ≥ 9. (2.1) X I(X) i i=1 2.3. CONNECTIVITY OF MAXIMAL PLANAR GRAPHS 17

Figure 2.10: A maximal planar graph with δ(G) = 5 and ∆(G) = 6.

Let G0 denote the planar graph induced by X ∪ I(X). Since one more edge can be added to G0 to obtain a maximal planar graph, it follows from Theorem 1.1 of Euler and Equation 2.1 that d(v) ≤ 6|I(X)| − 7. X v∈I(X) The desired result for the interior follows directly from this last inequality. As X is the vertex set of an arbitrary separating 4–cycle, the result holds also for the exterior of X.

2.3 Connectivity of maximal planar graphs

In this section we turn towards the connectivity of maximal planar graphs. Note that there is only one 2–connected maximal planar graph on three vertices, see Figure 2.11. As from now we consider only maximal planar graphs on n ≥ 4 vertices. First we will consider the connectivity of maximal planar graphs with minimum degree three.

Theorem 2.28 Let G be a maximal planar graph with minimum degree three. Then G is 3–connected.

Proof. Since G is maximal planar the boundary of every face is a 3–cycle. Therefore, G is at least 2–connected. G is at most 3–connected, since the minimum degree is three. 18 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

Figure 2.11: A 2–connected maximal planar graph with three vertices.

Assume G is not 3–connected. There exists a separating 2–cut with vertices u, v ∈ V (G). If uv ∈ E(G), then N(u) ∩ N(v) =6 ∅. Hence, G − {u, v} is connected. If uv ∈/ E(G), then N(u) ∩ N(v) =6 ∅ or N(u) ∩ N(v) = ∅ but in both cases we get G − {u, v} is connected.

We proceed in proving the connectivity of maximal planar graphs with minimum degree four. Theorem 2.29 Let G be a maximal planar graph with minimum degree four. Then G is either 4–connected or G has a separating triangle. Proof. Since G is a maximal planar graph the boundary of every face is a 3–cycle. There- fore, G is at least 2–connected. G is at most 4–connected, since the minimum degree is four. Assume there exists a separating 2–cut with vertices u, v ∈ V (G). If uv ∈ E(G), then N(u)∩N(v) =6 ∅. Hence, G−{u, v} is connected. If uv ∈/ E(G), then N(u)∩N(v) =6 ∅ or N(u) ∩ N(v) = ∅ but in both cases we get G − {u, v} is connected. Therefore, if G has no separating triangle and no separating 4–cycle then G is 5–connected. If G is not 5–connected, then there exists a separating triangle or a separating 4–cycle.

This theorem has an easy corollary. Corollary 2.30 Let G be a maximal planar graph with minimum degree four and without a separating triangle. Then G is 4–connected. Note that for a maximal planar graph, 3–cuts correspond to separating triangles, while 4–cuts correspond to separating 4–cycles. Theorem 2.31 Let G be a maximal planar graph with minimum degree ﬁve. Then G is either 5–connected or G has a separating triangle or a separating 4–cycle. Proof. Since G is a maximal planar graph the boundary of every face is a 3–cycle. Therefore, G is at least 2–connected. G is at most 5–connected, since the minimum degree is ﬁve. Assume there exists a separating 2–cut with vertices u, v ∈ V (G). If uv ∈ E(G), then N(u) ∩ N(v) =6 ∅. Hence, G − {u, v} is connected. If uv ∈/ E(G), then N(u) ∩ N(v) =6 ∅ or N(u) ∩ N(v) = ∅ but in both cases we get G − {u, v} is connected. Therefore, if G has no separating triangle then G is 4–connected. If G is not 4–connected, then there exists a separating triangle. 2.3. CONNECTIVITY OF MAXIMAL PLANAR GRAPHS 19

This theorem yields the following two corollaries.

Corollary 2.32 Let G be a maximal planar graph with minimum degree ﬁve and without a separating triangle. Then G is 4–connected.

Corollary 2.33 Let G be a maximal planar graph with minimum degree ﬁve and without a separating triangle and without a separating 4–cycle. Then G is 5–connected.

Now consider a maximal planar graph with vertex degrees d1 ≥ d2 ≥ ... ≥ dn. Re- member τi(G) denotes the number of all vertices with degree i. Our purpose is to give suﬃcient conditions on d1 ≥ d2 ≥ ... ≥ dn for G to be dn–connected. If dn = 3, G will always be dn–connected. Hence, we assume that 4 ≤ dn ≤ 5. In 1978 Hakimi and Schmeichel proved the following result.

Theorem 2.34 (Hakimi and Schmeichel [30]) Let G be a maximal planar graph with vertex degrees d1 ≥ d2 ≥ ... ≥ dn, with dn ≥ 4. If 7 τ (G) + τ (G) < 14, j4 4 k 5 then G is dn–connected.

One of the most elementary properties that a graph can have is that of being connected. This is a global property in the sense that it is deﬁned in terms of all pairs of vertices in the graph. Now we will present a result which uses the property of being connected in a localized sense.

Deﬁnition 2.35 A graph G is called locally k–connected with k ∈ IN if for each vertex v, the subgraph induced by N(v) is k–connected.

We obtain the following lemma.

Lemma 2.36 Each maximal planar graph G = (V, E) with |V | ≥ 4 is locally 2–connected.

Proof. Let G be a maximal planar graph and v ∈ V (G) a vertex of G. The subgraph induced by N(v) is a cycle with or without chords and thus 2–connected.

Thus we get the next corollary.

Corollary 2.37 If v is a vertex of a locally 2–connected planar graph, then v belongs to exactly d(v) triangles.

It is well known that every 6–connected graph is non-planar, while there are 5– connected graphs which are planar. For local connectedness, however, the situation is quite diﬀerent.

Lemma 2.38 Every locally 3–connected graph G is non-planar. 20 CHAPTER 2. FUNDAMENTALS OF MAXIMAL PLANAR GRAPHS

Proof. Let G be locally 3–connected and v be a vertex of G. We consider two cases.

Case 1: N(v) is complete. Since N(v) is 3–connected for each v ∈ V (G), N(v) contains the complete graph K4 as a subgraph. In G, the vertex v is adjacent to all vertices of K4 so that G contains the K5 as a subgraph, and by Kuratowski’s Theorem 1.3, G is non-planar.

Case 2: N(v) is not complete. Therefore, there exist two non-adjacent vertices u and w and three disjoint P (u, w) paths, each of which has length at least two. In G, the vertex v is adjacent to the interior vertices of these three paths. Thus G contains a subgraph homeomorphic with K3,3 and by Kuratowski’s Theorem 1.3, G is non-planar. Chapter 3

Hamiltonicity of maximal planar graphs

The hamiltonian problem; determining when a graph G contains a cycle, which contains all vertices of G exactly once, has been fundamental in graph theory. Named after Sir William Rowan Hamilton, this problem has its origins in the 1850s. It was the Four Color Problem which stimulated interest in hamiltonian planar graphs; if a planar cubic graph is hamiltonian then it is easy to color its faces by four colors in such a way that adjacent faces always receive distinct colors.

3.1 Hamiltonian cycle

It is well known that the problem of recognizing hamiltonian graphs is NP-complete. The problem does not get any easier when the input is restricted to 3–connected planar graphs, as proved by M. R. Garey, D. S. Johanson and R. E. Tarjan [29]. In 1980 T. Akiyama, T. Nishizeki and T. Saito [1] modified the argument to show that the problem remains NP- complete even if the input is restricted to cubic bipartite planar graphs. In particular, V. Chvátal and A. Wigderson showed independently that the problem of determining whether a maximal planar graph is hamiltonian is NP-complete, see V. Chvátal [20]. However, H. Whitney proved that any maximal planar graph without separating triangles is hamiltonian. Since Whitney’s result plays a decisive role, we state this result first. For Whitney’s Theorem, we need the following definition and the next lemma. H. Whitney introduced the following condition.

Definition 3.1 Let G be a planar triangulation, let R be the exterior face of G, and let a and b be two vertices on R. We say that (G, R, a, b) satisfies Whitney’s Condition (Condition (W) for short) if (G, R, a, b) satisfies Condition (W1) and (W2) described below. We say that (G, R, a, b) satisfies Condition (W1) if G has no separating triangles. We say that (G, R, a, b) satisfies Condition (W2) if either (G, R, a, b) satisfies Condition (W2a) R is divided into two paths, a0a1...am is the path from a to b and b0b1...bn is the path from b to a (a0 = bn = a, b0 = am = b), and there is no chord of the form aiaj or bibj, see Figure 3.1, or

21 22 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Condition (W2b) R is divided into three paths, a0a1...am is the path from a to b, b0b1...bn is the path from b to c and c0c1...ck is the path from c to a (a0 = ck = a, b0 = am = b, c0 = bn = c) for some vertex c on R distinct from a and b and there is no chord of the form aiaj, bibj or cicj, see Figure 3.1.

In 1931 H. Whitney proved the following.

Lemma 3.2 (Whitney [55]) Let G be a planar triangulation, let R be the exterior face of G, and let a and b be two vertices on R. If (G, R, a, b) satisﬁes Condition (W), then G has a hamiltonian path from a to b.

a a1 a a1

b4 a2 c2 a2

b3 b c1 b

b2 b1 c b1 (W2a) (W2b) Figure 3.1: An illustration of conditions (W2a) and (W2b).

Theorem 3.3 (Whitney [55]) Let G be a maximal planar graph without separating triangles. Then G is hamiltonian.

Tutte [53] generalized Whitney’s result.

Theorem 3.4 (Tutte [53]) Let G be a 4–connected planar graph. Then G is hamiltonian.

He proved this by showing that every planar graph has a special kind of path. This will be called a Tutte path, and is a generalization of a hamiltonian path. It is convenient to consider only 2–connected graphs.

Deﬁnition 3.5 For a subgraph H of a graph G, a trivial bridge and a non-trivial bridge of H in G are deﬁned as follows: A trivial bridge of H in G is an edge in E(G) \ E(H) with both ends in V (H). A non-trivial bridge of H in G is a component K of G − V (H) together with all vertices of H adjacent to vertices of K and all edges with one end in H and the other in K. The vertices of attachment of a bridge B of H in G are V (B)∩V (H). A bridge is attached to its vertices of attachment. A path (cycle) P , as a subgraph of a planar graph G, is a Tutte path (cycle) if and only if each bridge of P has at most three vertices of attachment and each bridge containing an edge of XG has at most two vertices of attachment. 3.1. HAMILTONIAN CYCLE 23

Lemma 3.6 (Tutte [53]) Let G be a 2–connected planar graph. Let x, y and α be two vertices and an edge, respectively, of XG. Then G has a Tutte path P from x to y containing α.

Tutte’s Theorem follows by choosing x and y to be adjacent. In 1983, C. Thomassen [51] improved the result of W. T. Tutte by removing the restriction on the location of y.

Lemma 3.7 (Thomassen [51]) Let G be a 2–connected planar graph. Let x and α be a vertex and an edge, respectively, of XG and let y be any vertex of G distinct from x. Then G has a Tutte path P from x to y containing α.

For a proof of Lemma 3.7 see [18]. Note that a Tutte path in a 4–connected graph is also a hamiltonian path. This provides the next corollary.

Corollary 3.8 (Thomassen [51]) Every 4–connected planar graph has a hamiltonian cycle through any edge of G.

Moreover, Lemma 3.7 allows a more generalized result. Therefore, we need the next deﬁnition.

Deﬁnition 3.9 A graph G is called hamiltonian–connected if it contains a hamiltonian path between any two prescribed vertices.

It is well known that hamiltonian–connectedness implies hamiltonicity. Thus the following corollary that follows from Lemma 3.7 is a generalization of Tutte’s result.

Corollary 3.10 (Thomassen [51]) Let G be a 4–connected planar graph. Then G is hamiltonian–connected.

D. P. Sanders [45] improved the result of C. Thomassen by removing the restriction on the location of x. He proved the following variation of Tutte’s Lemma 3.6.

Theorem 3.11 (Sanders [45]) Let G be a 2–connected planar graph. Let α be an edge of XG, and let x and y be arbitrary distinct vertices of G. Then G has a Tutte path P from x to y containing α.

The previous theorem leads to the next corollary that improves the result of C. Thomassen.

Corollary 3.12 (Sanders [45]) Let G be a 4–connected planar graph. Let α be an edge of XG, and let x and y be arbitrary distinct vertices of G. Then G has a hamiltonian path P from x to y containing α.

Note that Tutte’s Theorem shows that a 4–connected planar graph has a hamiltonian cycle and while Thomassen’s result shows that a 4–connected planar graph has a hamiltonian cycle through any edge, this corollary shows that a 4–connected planar graph has a hamiltonian cycle through any two edges. 24 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Corollary 3.13 (Sanders [45]) Every 4–connected planar graph has a hamiltonian cycle through any two of its edges.

In 2003 C. Chen [17] generalized Whitney’s result in a diﬀerent direction, he proved the following variation of Whitney’s lemma.

Lemma 3.14 (Chen [17]) Let G be a maximal planar graph without separating triangles, let R be the exterior face of G, and let a, b and c be the three vertices on R. Then G has a hamiltonian path from a to b starting from the edge ac.

From the Lemma above, C. Chen obtained a stronger version of Whitney’s Theorem.

Theorem 3.15 (Chen [17]) Let G be a maximal planar graph with only one separating triangle. Then G is hamiltonian.

While trying to answer the question of what makes a maximal planar graph with separating triangles non-hamiltonian, C. Chen introduced the deﬁnition of hamiltonian for any two boundary edges. From Lemma 3.14 above, C. Chen obtained a stronger version of Whitney’s Theorem.

Theorem 3.16 (Chen [17]) Let G be a maximal planar graph without separating triangles. Then G is hamiltonian for any two boundary edges.

With the following theorem we extend Theorem 3.16 to exactly one separating triangle.

Theorem 3.17 (Helden [33]) Let G be a maximal planar graph with only one separating triangle. Then G is hamiltonian for any two boundary edges.

Proof. Let x, y and z be the vertices of the separating triangle T and let xy be no boundary edge. Let Gin (Gout, respectively) be the subgraph of G derived by deleting all the vertices outside (inside, respectively) the separating triangle T . Gout and Gin are both maximal planar graphs without separating triangles. By Theorem 3.16 the graph Gin is hamiltonian for any two boundary edges. Consider Gout, let a, b and c be the vertices which form the exterior face of Gout. Let a be a vertex which is distinct from the vertices x 0 and y. Furthermore, let Gout be the subgraph of Gout derived by deleting a. By Theorem 0 3.11 Gout has a Tutte path P from x to y containing bc. Thus we obtain a Tutte path P in Gout from x to y containing ba and ac. Since Gout is a maximal planar graph without separating triangles, Gout is 4–connected. Note that a Tutte path in a 4–connected graph is also a hamiltonian path. Since Gin is hamiltonian for any two boundary edges, we can always ﬁnd a path Pin(x, y). Then acb...xPin(x, y)y...a is a hamiltonian cycle in G passing through two boundary edges. Since a is arbitrary G is hamiltonian for any two boundary edges.

With Theorem 3.16 and Theorem 3.17 we can prove the following result.

Theorem 3.18 (Helden [33]) Let G be a maximal planar graph with exactly two separating triangles. Then G is hamiltonian. 3.1. HAMILTONIAN CYCLE 25

Proof. Let a, b and c be the vertices of one separating triangle T . Let Gin (Gout, respectively) be the subgraph of G derived by deleting all the vertices outside (inside, respectively) the separating triangle T . We distinguish the two cases in which the two separating triangles are either nested or not.

Case 1: The two separating triangles are not nested. The two separating triangles could be disjoint or not. In both cases Gin is a graph without separating triangles and Gout is a graph with only one separating triangle. By Theorem 3.16 the graph Gin is hamiltonian for any two boundary edges. Note that a planar graph can always be embedded in the plane such that a given face of the graph becomes the exterior face, see Lemma 1.10. Therefore, we can embed Gout in the plane such that abca becomes the exterior cycle of Gout. By Theorem 3.17 Gout is also hamiltonian for any two boundary edges. Then acPin(c, b)bPout(b, a)a is a hamiltonian cycle in G. Since Gin and Gout are hamiltonian for any two boundary edges, we can always ﬁnd one common edge of both paths Pin and Pout.

Case 2: The two separating triangles are nested. The two separating triangles could be disjoint or not. In both cases we have to distinguish the following two cases.

Subcase 2.1: Gin is a graph without separating triangles and Gout is a graph with only one separating triangle.

Subcase 2.2: Gin is a graph with only one separating triangle and Gout is a graph without separating triangles. In both Subcases we obtain that Gin is hamiltonian for any two boundary edges and Gout is hamiltonian for any two boundary edges. Then acPin(c, b)bPout(b, a)a is a hamiltonian cycle in G.

Now we describe how to decompose a maximal planar graph G with at least one separating triangle into at least two maximal planar graphs with fewer separating triangles than G.

Deﬁnition 3.19 Let G be a maximal planar graph and let D1, D2, ..., Dk be k separating triangles. Let Gin,D1 (Gin,D2 , ..., Gin,Dk , respectively) be the subgraph of G derived by deleting all the vertices outside the separating triangle D1 (D2, ..., Dk, respectively). Let

Gout,D1,D2,...,Dk be the subgraph of G derived by deleting all the vertices inside the separating triangles D1, D2, ..., Dk.

Note that the graphs Gin,D1 (Gin,D2 , ..., Gin,Dk , respectively) and Gout,D1,D2,...,Dk are maximal planar graphs with fewer separating triangles than G. The following lemma shows how to obtain a hamiltonian cycle in G from two known hamiltonian cycles in these subgraphs.

Lemma 3.20 (Helden and Vieten [34]) Let G be a maximal planar graph with at least one separating triangle D. If Gin,D is hamiltonian for any two boundary edges of Gin,D and Gout,D is hamiltonian for any two boundary edges of Gout,D, then G is hamiltonian. 26 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Proof. Let a, b and c be the vertices of the separating triangle D. Consider Gin,D. a, b and c are the vertices of the exterior cycle of Gin,D. Without loss of generality, Gin,D has a hamiltonian cycle Cin = abcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − b between c and a. Next consider Gout,D. a, b and c are the vertices of a facial cycle of Gout,D. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. Without loss of generality, Gout,D has a hamiltonian cycle Cout = bcaPout(a, b)b where Pout(a, b) is a hamiltonian path of Gout,D − c between a and b. Then C = aPout(a, b)bcPin(c, a)a is a hamiltonian cycle of G.

In 2002 B. Jackson and X. Yu improved Corollary 3.13 of D.P. Sanders in a diﬀerent direction. They showed the following.

Theorem 3.21 (Jackson and Yu [37]) Let G be a planar triangulation without separating triangles. Let T , T1 and T2 be distinct triangles in G. Let V (T ) = {u, v, w}. Then there exists a hamiltonian cycle C of G and edges e1 ∈ E(T1) and e2 ∈ E(T2) such that uv, uw, e1, e2 are distinct and contained in E(C).

This implies the following two easy corollaries.

Corollary 3.22 (Helden and Vieten [34]) Let G be a maximal planar graph without separating triangles. Let T1 and T2 be distinct triangles in G. Let V (XG) = {u, v, w}. Then there exists a hamiltonian cycle C of G and edges e1 ∈ E(T1) and e2 ∈ E(T2) such that uv, uw, e1, e2 are distinct and contained in E(C).

Corollary 3.23 (Helden and Vieten [34]) Let G be a maximal planar graph without separating triangles. Let T1 and T2 be distinct, arbitrary and facial triangles in G. Then there exist edges e1 ∈ E(T1) and e2 ∈ E(T2) such that G is hamiltonian for any two boundary edges of G and the hamiltonian cycle contains the edges e1 and e2.

With Corollary 3.22 we obtain a stronger version of Theorem 3.16 of C. Chen.

Corollary 3.24 (Helden and Vieten [34]) Let G be a maximal planar graph without separating triangles. Let T be an arbitrary facial triangle in G. Then there exists an edge e ∈ E(T ) such that G is hamiltonian for any two boundary edges of G and the hamiltonian cycle contains the edge e.

Let G be a maximal planar graph. Suppose that T is a separating triangle in G. Then we may separate G into two graphs Gin,T and Gout,T . Then Gin,T and Gout,T are maximal planar graphs and T is a facial cycle of Gin,T and Gout,T . We shall refer to T as a marker triangle in Gin,T and Gout,T . We now iterate this procedure for both Gin,T and Gout,T . We continue until we obtain a collection S of maximal planar graphs each of which has no separating triangles. We shall refer to S as pieces of G. Note that each separating triangle will occur as a marker triangle in exactly two pieces of G. We define a new graph B whose vertices are the pieces in S, and in which two pieces are joined by an edge if they have a marker triangle in common. It follows from the decomposition theory 3.1. HAMILTONIAN CYCLE 27 developed by W. H. Cunningham and J. Edmonds [22] that B is a tree and also that the set S and the tree B are uniquely defined by G. We shall refer to B as the decomposition tree of G. For a given embedding of G with k separating triangles D1, D2, ..., Dk we define the piece Gout,D1,D2,...,Dk as the root of the decomposition tree B. Thus we get a rooted decomposition tree B . For the proof of Theorem 3.27, we need the following two lemmata. Lemma 3.25 (Helden and Vieten [34]) Let G be a maximal planar graph with at least one separating triangle D1 and let B be the rooted decomposition tree of the given embedding of

G. Let each vertex of the tree B have at most two children and the root Gout,D1,D2,...,Dk has exactly one child Gin,D1 . If Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 , then G is hamiltonian for any two boundary edges of G for the given embedding of G.

Proof. Let a, b and c be the vertices of the separating triangle D1. Consider Gin,D1 .

Let a, b and c be the vertices of the exterior cycle of Gin,D1 . Without loss of generality,

Gin,D1 has a hamiltonian cycle Cin = abcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D1 − b between c and a.

Next consider Gout,D1 . Gout,D1 is a maximal planar graph without separating triangles. a, b and c are the vertices of the facial cycle Gout,D1 . By Corollary 3.24 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 and contains without loss of generality the edge ac. We replace the edge ac by the path Pin(c, a) in the graph G. Then G is hamiltonian for any two boundary edges of G for the given embedding of G.

Lemma 3.26 (Helden and Vieten [34]) Let G be a maximal planar graph with at least two separating triangles D1 and D2 and let B be the rooted decomposition tree of the given embedding of G. Let each vertex of the tree B have at most two children and the root

Gout,D1,D2,...,Dk has exactly two children Gin,D1 and Gin,D2 . If Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 and Gin,D2 is hamiltonian for any two boundary edges of Gin,D2 , then G is hamiltonian for any two boundary edges of G for the given embedding of G.

Proof. Let a1, b1 and c1 be the vertices of the separating triangle D1. Consider Gin,D1 . a1, b1 and c1 are the vertices of the exterior cycle of Gin,D1 . Without loss of generality,

Gin,D1 has a hamiltonian cycle Cin,D1 = a1b1c1Pin,D1 (c1, a1)a1 where Pin,D1 (c1, a1) is a hamiltonian path of Gin,D1 − b1 between c1 and a1.

Let a2, b2 and c2 be the vertices of the separating triangle D2. Consider Gin,D2 . a2, b2 and c2 are the vertices of the exterior cycle of Gin,D2 . Without loss of generality, Gin,D2 has a hamiltonian cycle Cin,D2 = a2b2c2Pin,D2 (c2, a2)a2 where Pin,D2 (c2, a2) is a hamiltonian path of Gin,D2 − b2 between c2 and a2.

Next consider Gout,D1,D2 . Gout,D1,D2 is a maximal planar graph without separating triangles. a1, b1, c1 and a2, b2, c2 are the vertices of distinct facial cycles of Gout,D1,D2 . By

Corollary 3.23 Gout,D1,D2 is hamiltonian for any two boundary edges of Gout,D1,D2 and contains without loss of generality the edges a1c1 and a2c2. We replace the edge a1c1 by the path Pin,D1 (c1, a1) and the edge a2c2 by the path Pin,D2 (c2, a2) in the graph G. Then G is hamiltonian for any two boundary edges of G for the given embedding of G. 28 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

From Lemma 3.25 and Lemma 3.26 above we obtain a stronger version of Theorem 3.16 and Theorem 3.17.

Theorem 3.27 (Helden and Vieten [34]) If G is a maximal planar graph with exactly two separating triangles, then G is hamiltonian for any two boundary edges of G.

Proof. Let D1 and D2 be the separating triangles of G. Let B be the rooted decomposition tree of an arbitrary embedding of G. Each vertex of the tree B has at most two children and the root Gout,D1,D2 has at least one child. We distinguish the following two cases.

Case 1: The root Gout,D1,D2 has exactly one child Gin,D1 . Then the piece Gin,D2 is a child of the piece Gin,D1 . The graph Gin,D1 has one separating triangle D2. By Theorem

3.17 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 . By Lemma 3.25 G is hamiltonian for any two boundary edges of G.

Case 2: The root Gout,D1,D2 has exactly two children Gin,D1 and Gin,D2 . The graph

Gin,D1 has no separating triangles and the graph Gin,D2 has no separating triangles. Then by Theorem 3.16 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 and Gin,D2 is hamiltonian for any two boundary edges of Gin,D2 . By Lemma 3.26 G is hamiltonian for any two boundary edges of G.

The hypothesis of Theorem 3.27 that G has at most two separating triangles cannot be weakened to at most three separating triangles. To show this we consider the graph G obtained from K4 by inserting a new vertex into each interior face of K4 then joining the new vertex to every vertex incident with the face. Then G is not immediately hamiltonian for any two boundary edges of G.

The following theorem presents one of our main results, because we do not claim an integer which is an upper bound on the number of separating triangles, but we assume a special structure of the position of the separating triangles to each other.

Theorem 3.28 (Helden and Vieten [34]) Let G be a maximal planar graph and let B be the rooted decomposition tree of the given embedding of G. If each vertex of the tree B has at most two children, then G is hamiltonian for any two boundary edges of G for the given embedding of G.

Proof. The proof is by induction on the number s of separating triangles. Clearly, the theorem is true for s = 0, s = 1 and s = 2. The claim follows from Theorem 3.16, The- orem 3.17 and Theorem 3.27. We proceed with the induction step. Assume that G has s+1 separating triangles. The root Gout,D1,D2,...,Ds+1 has at least one child. We distinguish the following two cases.

Case 1: The root Gout,D1,D2,...,Ds+1 has exactly one child Gin,D1 . The graph Gin,D1 has s separating triangles. Since the rooted decomposition tree T of Gin,D1 is a subtree of the tree B, each vertex of the tree T has at most two children. Then by induction Gin,D1 is 3.1. HAMILTONIAN CYCLE 29

hamiltonian for any two boundary edges of Gin,D1 . By Lemma 3.25 G is hamiltonian for any two boundary edges of G for the given embedding of G.

Case 2: The root Gout,D1,D2,...,Ds+1 has exactly two children Gin,D1 and Gin,D2 .The graph Gin,D1 has t separating triangles and the graph Gin,D2 has r separating triangles. Since r + t = s − 1 it is necessary that t ≤ s and r ≤ s. Since the rooted decomposition tree T1 of Gin,D1 and the rooted decomposition tree T2 of Gin,D2 are subtrees of the tree

B, each vertex of this subtrees has at most two children. Then by induction Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 and Gin,D2 is hamiltonian for any two boundary edges of Gin,D2 . By Lemma 3.26 G is hamiltonian for any two boundary edges of G for the given embedding of G.

With Theorem 3.27 we can extend Theorem 3.18 to exactly three separating triangles.

Theorem 3.29 (Helden and Vieten [34]) If G is a maximal planar graph with exactly three separating triangles, then G is hamiltonian.

Proof. Let D1, D2 and D3 be the separating triangles of G. Let B be the rooted decomposition tree of an arbitrary embedding of G. Each vertex of the tree B has at most three children and the root Gout,D1,D2,D3 has at least one child. We distinguish the following three cases.

Case 1: The root Gout,D1,D2,D3 has exactly one child Gin,D1 . Then the graph Gin,D1 has two separating triangles D2 and D3. By Theorem 3.27 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 . The graph Gout,D1 has no separating triangles. Then by

Theorem 3.16 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . By Lemma 3.20 G is hamiltonian.

Case 2: The root Gout,D1,D2,D3 has exactly two children Gin,D1 and Gin,D2 . Without loss of generality, the graph Gin,D1 has one separating triangle D3. Then the graph Gout,D1 has also one separating triangle D2. Then by Theorem 3.17 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 and Gout,D1 is hamiltonian for any two boundary edges of

Gin,D2 . By Lemma 3.20 G is hamiltonian.

Case 3: The root Gout,D1,D2,D3 has exactly three children Gin,D1 , Gin,D2 and Gin,D3 .

The graph Gin,D1 has no separating triangles and the graph Gout,D1 has two separating triangles D2 and D3. Then by Theorem 3.16 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 and by Theorem 3.27 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . By Lemma 3.20 G is hamiltonian.

Analogously to Theorem 3.29 we can prove the unbounded version with Theorem 3.28. All at once this is also a generalization of Theorem 3.28.

Theorem 3.30 (Helden and Vieten [34]) Let G be a maximal planar graph and let B be the rooted decomposition tree of the given embedding of G. If the root of the tree B has 30 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS exactly three children and have all other vertices of the tree B at most two children, then G is hamiltonian.

Proof. Let B be the rooted decomposition tree of the given embedding of G. Let

Gin,D1 , Gin,D2 and Gin,D3 be the three children of the root of the tree B. Since the rooted decomposition tree T1 of Gin,D1 and the rooted decomposition tree T2 of Gout,D1 are subtrees of the tree B, each vertex of these subtrees has at most two children. By Theorem

3.28 the graph Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 and the graph

Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . By Lemma 3.20 the graph G is hamiltonian.

The hypothesis of Theorem 3.28 that each vertex of the rooted decomposition tree B has at most two children cannot be weakened to at most three children. To show this we consider the graph G obtained from K4 by inserting a new vertex into each interior face of K4 then joining the new vertex to every vertex incident with the face. Then the rooted decomposition tree of G has exactly three children and G is not immediately hamiltonian for any two boundary edges of G. In the following we extend Theorem 3.27 to four separating triangles. The ﬁrst to take the position of a hamiltonian cycle into account were T. Asano, T. Nishizeki and T. Watanabe [4]. They introduced the following deﬁnition.

Deﬁnition 3.31 Let G be a hamiltonian maximal planar graph with a hamiltonian cycle C. We say a (triangular) face of G to be frontier (with respect to the hamiltonian cycle C) if it has one or two edges in common with C, or else non-frontier. A frontier face is called a 1-frontier face if the face has exactly one edge in common with C, or else a 2-frontier face. We deﬁne kN (C) as the number of non-frontier faces and k1F (C) and k2F (C) as the number of 1-frontier and 2-frontier faces, respectively.

Lemma 3.32 Let G be a hamiltonian maximal planar graph on n vertices. Then k + 2k n = 1F 2F and k = k − 4. 2 N 2F Proof. Let G = (V, E) with |V | = n and |E| = m be a hamiltonian maximal planar graph with a hamiltonian cycle C and let l(G) be the number of faces of G. Then The- orem 1.1 of Euler yields l(G) = m − n + 2. On the other side, counting the frontier and non-frontier faces yields l(G) = k1F + k2F + kN . Moreover, 3l(G) = 2m, since each edge is located on the boundary of two faces. Then with Theorem 1.1 of Euler we obtain that m = 3n−6. The equation of these formulas provides k1F +k2F +kN = 2n−4. Together with k1F +2k2F n = 2 we obtain that k1F +k2F +kN = k1F +2k2F −4 and this yields kN = k2F −4.

The previous lemma provides the following immediate corollary.

Corollary 3.33 If G is a hamiltonian maximal planar graph, then

k2F ≥ 4. 3.1. HAMILTONIAN CYCLE 31

For the proof of Theorem 3.36 we need the next lemma and the following deﬁnition.

Lemma 3.34 (Helden [35]) Let G = (V, E) be a maximal planar graph without separating triangles and |V | ≥ 7. Then G has at least two vertices x and y with dG(x) ≥ 5 and dG(y) ≥ 5 and |N(x) ∩ N(y)| ≥ 2.

Proof. If G is a maximal planar graph with δ(G) = 5, then it is obvious. Therefore, let G be a maximal planar graph with δ(G) = 4. We distinguish the following three cases.

Case 1: At least one component of the subgraph G [D4] is a trivial graph. By Corol- lary 2.22 the graph G has at least two vertices x and y with dG(x) ≥ 5 and dG(y) ≥ 5 and |N(x) ∩ N(y)| ≥ 2.

Case 2: At least one component of the subgraph G [D4] is a cycle Cp with p ≥ 5. By Lemma 2.21 G has exactly two vertices x and y with dG(x) ≥ 5 and dG(y) ≥ 5 and |N(x) ∩ N(y)| ≥ 2.

Case 3: At least one component of the subgraph G [D4] is a path of length q with q ∈ IN. By Lemma 2.18 G has at least two vertices x and y with dG(x) ≥ 5 and dG(y) ≥ 5 and |N(x) ∩ N(y)| ≥ 2.

Deﬁnition 3.35 Let G be a maximal planar graph without separating triangles and with n ≥ 7 vertices. Let x and y be two vertices with dG(x) ≥ 5 and dG(y) ≥ 5 and |N(x) ∩ N(y)| ≥ 2. Let a, b ∈ N(x) ∩ N(y) with ab is an edge of G. The operation T is deﬁned as follows: T contracts the two vertices a and b to one vertex z. The edges xa (ya) and xb (yb) are contracted to one edge xz (yz, respectively). The new graph G0 is a maximal planar graph without separating triangles and with n − 1 vertices, see Figure 3.2.

x x

a b z

y y

Figure 3.2: Operation T .

In general a planar graph G has many embeddings on a surface. We shall now deﬁne an equivalence relation among these embeddings. Two embeddings of a planar graph 32 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS are equivalent if the boundary of a face in one embedding always corresponds to the boundary of a face in the other. We say that the planar embedding of a graph is unique if the embeddings are all equivalent. H. Whitney [56] proved that the embedding of a 3–connected planar graph is unique.

Theorem 3.36 (Helden [35]) If G is a maximal planar graph without separating triangles and with n ≥ 7 vertices, then a hamiltonian cycle C of G exists so that k2F = 4 and at least two 2-frontier faces have one edge in common.

Proof. The proof is by induction on the number of vertices of G. Assume ﬁrst n = 7 in order to establish the inductive base. See Figure 3.3, the thick lines represent the hamiltonian cycle.

Figure 3.3: Inductive base of the proof of Theorem 3.36.

Thus we have shown that the theorem is true if n = 7. For the inductive step, we assume next that n ≥ 8 and the theorem is true for any maximal planar graph with less than n vertices. Suppose that G is a maximal planar graph without separating triangles and with n vertices. Lemma 3.34 yields that G has two vertices x and y with dG(x) ≥ 5 and dG(y) ≥ 5 and |N(x) ∩ N(y)| ≥ 2. Let a, b ∈ N(x) ∩ N(y) with ab is an edge of G. Now we can apply the operation T . T contracts the two vertices a and b to one vertex z. The edges xa (ya) and xb (yb) are contracted to one edge xz (yz, respectively). The new graph G0 is a maximal planar graph without separating triangles and with n − 1 vertices. Thus, we may apply induction to G0. Without loss of generality, there exists a 0 0 0 hamiltonian cycle C of G such that k2F (C ) = 4 and at least two 2-frontier faces have one edge in common. Now we have to prove that there exists a hamiltonian cycle C of G such that k2F (C) = 4 and at least two 2-frontier faces have one edge in common. Let u be the vertex of G which is adjacent to the vertices x and a and let v be the vertex of G which is adjacent to the vertices x and b. The vertices u, x and z are the vertices of a facial cycle 0 0 F1 in G and the vertices v, z and x are the vertices of a facial cycle F2 in G . Both facial cycles have the edge xz in common. We know G0 has a hamiltonian cycle C0 such that at least two 2-frontier faces have one edge in common. Note that a planar graph can always be embedded on a surface such that a given face of the graph becomes the exterior face, 3.1. HAMILTONIAN CYCLE 33

x x

T −1 F1 F2

u a b v u z v Figure 3.4: Operation T −1. see Chiba and Nishizeki [19]. Since G0 is at least 3−connected the embedding of G0 is unique. Without loss of generality, we can embed the graph G0 on a surface such that the two 2-frontier faces with one edge in common have the edge xz in common. Note, if we determine one face, we can always ﬁnd at most three hamiltonian cycles, such that each edge of the given face can become the common edge of the 2-frontier faces with one edge in common for the corresponding hamiltonian cycle. This means F1 and F2 are the two 2-frontier faces in this embedding of G0. We distinguish the following two cases.

Case 1: uxzv is a path of the hamiltonian cycle C0. Now we can apply the operation T −1, see Figure 3.4 T −1 extracts the vertex z to the two vertices a and b. The edge xz is extracted to the two edges xa and xb. Then uxabv is a path of the hamiltonian cycle C of G. The vertices u, x and a are the vertices of a facial cycle K1 in G and the vertices x, a and b are the vertices of a facial cycle K2 in G. Both facial cycles have the edge xa in common. Therefore, K1 and K2 are two 2-frontier faces with one edge in common in an embedding of G. The two new facial cycles with vertices x, b, v and a, b, y are both 1-frontier faces with respect to the hamiltonian cycle C.

Case 2: uzxv is a path of the hamiltonian cycle C0. Now we can also apply the operation T −1. T −1 extracts the vertex z to the two vertices a and b. The edge xz is extracted to the two edges xa and xb. Then uabxv is a path of the hamiltonian cycle C of G. The vertices a, b and x are the vertices of a facial cycle K1 in G and the vertices b, x and v are the vertices of a facial cycle K2 in G. Both facial cycles have the edge bx in common. Therefore, K1 and K2 are two 2-frontier faces with one edge in common in an embedding of G. The two new facial cycles with vertices u, a, x and a, b, y are both 1-frontier faces with respect to the hamiltonian cycle C.

Let G be a hamiltonian maximal planar graph without separating triangles. By The- orem 3.36 there exists a hamiltonian cycle C of G such that k2F (C) = 4 and at least two 2-frontier faces have one edge in common. Let G1 be the graph obtained from G by inserting a new vertex into a facial cycle F of G then joining the new vertex to every vertex a, b and c incident with the face F . G1 is a hamiltonian maximal planar graph with one separating triangle. Now we can extend the hamiltonian cycle C of G to a hamiltonian cycle C1 of G1. We replace an edge of the facial cycle F which belongs to the hamiltonian cycle C by two edges which join the new vertex to the hamiltonian cycle C. 34 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

If F is a 2-frontier face with respect to the hamiltonian cycle C, which has no edge in common with another 2-frontier face, then k2F (C1) = 5. If F is a 2-frontier face with respect to the hamiltonian cycle C, which has an edge in common with another 2-frontier face, then k2F (C1) = 4. If F is a 1-frontier face with respect to the hamiltonian cycle C, then k2F (C1) = 5. Thus we get, k2F (C1) is increased by at most one if we add a separating triangle with one vertex. By Lemma 3.32 kN (C1) is also increased by at most one. Since G is at least 3−connected the embedding of G is unique. Therefore, we can embed the graph G on a surface such that F is a 2-frontier face, which has no edge in common with another 2-frontier face. Then k2F (C1) = k2F (C) + 1 and kN (C1) = 1. By the same argument, we can embed the graph G on a surface such that F is a 2-frontier face, which has an edge in common with another 2-frontier face. Then k2F (C1) = 4 = k2F (C) and kN (C1) = 0.

Let G2 be the graph obtained from G by inserting two new vertices each into two distinct facial cycles F1 and F2 of G then joining the new vertex to every vertex incident with the faces F1 and F2 respectively. G2 is a hamiltonian maximal planar graph with two separating triangles. Note that there exists a hamiltonian cycle C of G such that k2F = 4 and at least two 2-frontier faces have one edge in common. We can extend the hamiltonian cycle C of G to a hamiltonian cycle C2 of G2. Since G is at least 3−connected the embedding of G is unique. Therefore, we can embed the graph G on a surface such that F1 and F2 are 2-frontier faces and at least one of these has an common edge with another 2-frontier face. Then 4 ≤ k2F (C2) ≤ k2F (C) + 1 = 5 and kN (C2) ≤ 1.

Let Gi be the graph obtained from G by inserting i new vertices each into i distinct facial cycles F1, F2, ..., Fi of G then joining the new vertex to every vertex incident with the faces Fi respectively. Gi is a hamiltonian maximal planar graph with i separating triangles and a hamiltonian cycle Ci. We can embed the graph G on a surface such that Fs and Ft with 1 ≤ s, t ≤ i are 2-frontier faces and at least one of these has an common edge with another 2-frontier face. Then k2F (Ci) ≤ 4+i−1 = i+3 and kN (Ci) ≤ i−1.

We can now prove a stronger version of Theorem 3.29. Theorem 3.37 (Helden [35]) If G is a maximal planar graph with exactly four separating triangles, then G is hamiltonian.

Proof. Let G be a maximal planar graph with exactly four separating triangles. D1, D2, D3 and D4 are the 4 separating triangles. Let Gin,D1 be the subgraph of G derived by deleting all the vertices outside the separating triangle D1. Let Gout,D1 be the subgraph of G derived by deleting all the vertices inside the separating triangle D1. We distinguish the following four cases.

Case 1: The graph Gin,D1 has two separating triangles and the graph Gout,D1 has one separating triangle. By Theorem 3.27 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 . By Theorem 3.17 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . By Lemma 3.20 G is hamiltonian. 3.1. HAMILTONIAN CYCLE 35

Case 2: The graph Gin,D1 has one separating triangle and the graph Gout,D1 has two separating triangles. By Theorem 3.17 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 . By Theorem 3.27 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . By Lemma 3.20 G is hamiltonian.

Case 3: The graph Gin,D1 has no separating triangle and the graph Gout,D1 has three separating triangles. By Theorem 3.16 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 . By Theorem 3.29 Gout,D1 is hamiltonian. Let the vertices a, b and c be the vertices of the facial cycle D1 in Gout,D1 . The graph Gout,D1,D2,D3,D4 is a subgraph of Gout,D1 and has no separating triangles. If Gout,D1,D2,D3,D4 has n < 7 vertices, then there exists a hamiltonian cycle C of Gout,D1,D2,D3,D4 such that the hamiltonian cycle passes through at least one edge of the edges ab, bc and ca. If Gout,D1,D2,D3,D4 has n ≥ 7 vertices, then by Theorem 3.36 there exists a hamiltonian cycle C of Gout,D1,D2,D3,D4 such that k2F = 4 and at least two 2-frontier faces have one edge in common. Now we have to prove that the hamiltonian cycle C passes through at least one edge of the edges ab, bc and ca. We insert 3 new vertices each into the facial cycles D2, D3 and D4 of G then joining the new vertex to every vertex incident with the faces of D2, D3 and D4 respectively. The new graph H is a hamiltonian maximal planar graph with three separating triangles. Now we 0 can extend the hamiltonian cycle C of Gout,D1,D2,D3,D4 to a hamiltonian cycle C of H. 0 0 0 Thus we get k2F (C ) = 6 and kN (C ) = 2 with respect to the hamiltonian cycle C . Then we can embed the graph H on a surface such that D1 is not a non-frontier face. There- fore, the hamiltonian cycle H passes through at least one edge of the edges ab, bc and ca. At last we have to replace the vertices in the facial cycles D2, D3 and D4 by the graphs

Gin,D2 − D2, Gin,D3 − D3 and Gin,D4 − D4 respectively. Since Gin,Di with 2 ≤ i ≤ 4 are maximal planar graphs without separating triangles, the graphs Gin,Di with 2 ≤ i ≤ 4 are hamiltonian for any two boundary edges of Gin,Di . Therefore, the graphs Gin,Di − Di with

2 ≤ i ≤ 4 have a hamiltonian path in Gin,Di − Di. Now we replace the two edges which join the new vertices to the hamiltonian cycle H by the hamiltonian paths in Gin,Di − Di with 2 ≤ i ≤ 4. Thus we get a hamiltonian cycle H 0 which passes through at least one 0 edge of the edges ab, bc and ca. H is a hamiltonian cycle of Gout,D1 . Without loss of 0 generality, H passes through the edge ca. Consider Gin,D1 . a, b and c are the vertices of the exterior cycle of Gin,D1 . Without loss of generality, Gin,D1 has a hamiltonian cycle Cin = abcPin(c, a)a, where Pin(c, a) is a hamiltonian path of Gin,D − b between c and a. 0 Then we replace the edge ca in the hamiltonian cycle H of Gout,D1 by the hamiltonian path Pin(c, a) between c and a of Gin,D −b. Then the new cycle is a hamiltonian cycle of G.

Case 4: The graph Gin,D1 has three separating triangles and the graph Gout,D1 has no separating triangle. By Theorem 3.29 Gin,D1 is hamiltonian. By Theorem 3.16 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. Without loss of generality, Gout,D1 has a hamiltonian cycle passing through any two edges of the facial cycle D1. Then analogically to Case 3 G is hamiltonian. With Theorem 3.38 we can extend Theorem 3.37 to exactly ﬁve separating triangles. 36 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Theorem 3.38 (Helden [35]) If G is a maximal planar graph with exactly ﬁve separating triangles, then G is hamiltonian.

Proof. Let G be a maximal planar graph with exactly ﬁve separating triangles.

D1, D2, ..., D5 are the 5 separating triangles. Let Gin,D1 be the subgraph of G derived by deleting all the vertices outside the separating triangle D1. Let Gout,D1 be the subgraph of G derived by deleting all the vertices inside the separating triangle D1. We distinguish the following ﬁve cases.

Case 1: The graph Gin,D1 has three separating triangles and the graph Gout,D1 has one separating triangle. Then analogically to the proof of Theorem 3.37 Case 4 G is hamiltonian.

Case 2: The graph Gin,D1 has two separating triangles and the graph Gout,D1 has two separating triangles. Then analogically to the proof of Theorem 3.37 Case 1 G is hamiltonian.

Case 3:The graph Gin,D1 has one separating triangle and the graph Gout,D1 has three separating triangles. Then analogically to the proof of Theorem 3.37 Case 3 G is hamiltonian.

Case 4: The graph Gin,D1 has no separating triangle and the graph Gout,D1 has four separating triangles. By Theorem 3.16 Gin,D1 is hamiltonian for any two boundary edges of Gin,D1 . By Theorem 3.37 Gout,D1 is hamiltonian. Let the vertices a, b and c be the vertices of the facial cycle D1 in Gout,D1 . The graph Gout,D1,D2,...,D5 is a subgraph of Gout,D1 and has no separating triangles. If Gout,D1,D2,...,D5 has n < 7 vertices, then there exists a hamiltonian cycle C of Gout,D1,D2,...,D5 such that the hamiltonian cycle passes through at least one edge of the edges ab, bc and ca. If Gout,D1,D2,...,D5 has n ≥ 7 vertices, then by

Theorem 3.36 there exists a hamiltonian cycle C of Gout,D1,D2,...,D5 such that k2F = 4 and at least two 2-frontier faces have one edge in common. Now we have to prove that the hamiltonian cycle C passes through at least one edge of the edges ab, bc and ca. We insert 4 new vertices each into the facial cycles D2, D3, D4 and D5 of G then joining the new vertex to every vertex incident with the faces of D2, D3, D4 and D5 respectively. The new graph H is a hamiltonian maximal planar graph with 4 separating triangles. Now we can 0 extend the hamiltonian cycle C of Gout,D1,D2,...,D5 to a hamiltonian cycle C of H. Thus 0 0 0 we get k2F (C ) = 7 and kN (C ) = 3 with respect to the hamiltonian cycle C . Then we can embed the graph H on a surface such that D1 is not a non-frontier face. Therefore, the hamiltonian cycle C0 passes through at least one edge of the edges ab, bc and ca. At last we have to replace the vertices in the facial cycles D2, D3, D4 and D5 by the graphs

Gin,D2 − D2, Gin,D3 − D3, Gin,D4 − D4 and Gin,D5 − D5 respectively. Since Gin,Di with

2 ≤ i ≤ 5 are maximal planar graphs without separating triangles, the graphs Gin,Di with

2 ≤ i ≤ 5 are hamiltonian for any two boundary edges of Gin,Di . Therefore, the graphs

Gin,Di −Di with 2 ≤ i ≤ 5 have a hamiltonian path in Gin,Di −Di. Now we replace the two edges which join the new vertices to the hamiltonian cycle C 0 by the hamiltonian paths in 00 Gin,Di − Di with 2 ≤ i ≤ 5. Thus we get a hamiltonian cycle C which passes through at 3.1. HAMILTONIAN CYCLE 37

0 least one edge of the edges ab, bc and ca. H is a hamiltonian cycle of Gout,D1 . Without loss 00 of generality, C passes through the edge ca. Consider Gin,D1 . a, b and c are the vertices of the exterior cycle of Gin,D1 . Without loss of generality, Gin,D1 has a hamiltonian cycle

Cin = abcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D1 − b between c and a. 00 Then we replace the edge ca in the hamiltonian cycle C of Gout,D1 by the hamiltonian path Pin(c, a) between c and a of Gin,D1 −b. Then the new cycle is a hamiltonian cycle of G.

Case 5: The graph Gin,D1 has four separating triangles and the graph Gout,D1 has no separating triangle. By Theorem 3.37 Gin,D1 is hamiltonian. By Theorem 3.16 Gout,D1 is hamiltonian for any two boundary edges of Gout,D1 . Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Lemma 1.10. Without loss of generality, Gout,D1 has a hamiltonian cycle passing through any two edges of the facial cycle D1. Then analogically to Case 4 the graph G is hamiltonian.

Let G be a maximal planar graph with exactly six separating triangles. D1, D2, ..., D6 are the 6 separating triangles. Let Gin,D1 be the subgraph of G derived by deleting all the vertices outside the separating triangle D1. Let Gout,D1 be the subgraph of G derived by deleting all the vertices inside the separating triangle D1. Consider the case that the graph Gin,D1 has no separating triangles and the graph Gout,D1 has ﬁve separating triangles. Assume that Gout,D1,D2,...,D6 has n ≥ 7 vertices then by Theorem 3.36 there exists a hamiltonian cycle C of Gout,D1,D2,...,D6 such that k2F = 4 and at least two 2-frontier faces have one edge in common. Now we insert 5 new vertices each into the facial cycles D2, ..., D6 of G then joining the new vertex to every vertex incident with the faces of D2, ..., D6 respectively. The new graph H is a hamiltonian maximal planar graph with 5 separating triangles. Now we can extend the hamiltonian cycle C of Gout,D1,D2,...,D6 to a 0 0 0 hamiltonian cycle C of H. Thus we get k2F (C ) = 8 and kN (C ) = 4 with respect to the hamiltonian cycle C0. Consider the case that the three non-frontier faces have each one vertex in common. Then these three non-frontier faces determine a facial cycle which is the remaining non-frontier face with respect to the hamiltonian cycle C 0. This leads to a contradiction. Therefore, there must be a maximal planar graph with six separating triangles which is non-hamiltonian. Finally, if G is a maximal planar graph with exactly six separating triangles, then there is a counter-example which shows that G is not immediately hamiltonian, see Figure 3.5.

Deﬁnition 3.39 Let G = (V, E) be a simple graph. We call X ⊆ V (G) independent if E(G[X]) = ∅. A maximal independent vertex set is an independent set containing the largest possible number of vertices. Example 3.40 (Helden [35]) The graph from Figure 3.5 is a non-hamiltonian maximal planar graph. The vertices v5, v7, v11, v13, v12, v14, v15, v17 form a maximal independent vertex set. If a hamiltonian cycle C exists, then these 8 vertices cannot be adjacent. Hence, only two of the remaining 9 vertices can be adjacent to at most one vertex of the maximal independent vertex set. Since the vertices v1, v2 and v3 are adjacent to only one vertex of the maximal independent vertex set, there cannot exist a hamiltonian cycle. 38 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

v16 v17 v15 v6 v4 v12 v10 v14

v9 v11 v13

v1 v2

Figure 3.5: Counter-example: A non-hamiltonian maximal planar graph with exactly six

separating triangles.

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Figure 3.6: The smallest non-hamiltonian maximal planar graph with 11 vertices. 3.2. HAMILTONIAN PATH 39

Example 3.41 The graph G from Figure 3.6 is the smallest non-hamiltonian maximal planar graph with 11 vertices. The graph G has seven separating triangles. In 1980 T. Asano, T. Nishizeki and T. Watanabe [4] showed that every non-hamiltonian maximal planar graph with 11 vertices is isomorphic to the graph shown in Figure 3.6.

3.2 Hamiltonian path

Since maximal planar graphs with more than five separating triangles are not necessarily hamiltonian, we will show that maximal planar graphs with exactly six separating triangles have at least a hamiltonian path. Therefore, we need the following definition and the next lemma. Definition 3.42 Let G be a maximal planar graph and let a, b ∈ V (G) with ab ∈ E(G) and |N(a) ∩ N(b)| ≥ 2. Then we define the operation D, see Figure 3.7.

a a

x D

b b G G + x

Figure 3.7: Operation D.

Remark. After applying the operation D, the graph G + x is still a maximal planar graph. Lemma 3.43 Let G be a maximal planar graph with exactly one separating triangle. Let a, b and c be the vertices which form the separating triangle. Then we can apply the operation D to a, b ∈ V (G) and the graph G + x is a maximal planar graph without separating triangles. Proof. Since a, b and c form one separating triangle the vertices does not form the boundary of a face, especially not the exterior face. Because of the maximal planarity the vertices a and b have at least two common neighbors beside c, at least one vertex inside the separating triangle and one outside. Now we can apply the operation D. If we remove the vertices a, b and c, the graph G + x is still connected, because the new vertex connects the inside and the outside of the separating triangle. So the resulting graph is a maximal planar graph without separating triangle.

This lemma yields the next easy corollary. 40 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Corollary 3.44 Let G be a maximal planar graph with k > 1 separating triangles. Let a, b and c be the vertices which form one separating triangle. Then we can apply the operation D to a, b ∈ V (G) and the graph G + x is a maximal planar graph with at most k − 1 separating triangles. Remark. Let G be a maximal planar graph with two separating triangles. Let a, b and c be the vertices which form one separating triangle and let d, e and f be the vertices which form the other separating triangle. If the two separating triangles have one edge in common, say ab = de, then we can apply the operation D to a, b ∈ V (G) only once. The resulting graph G + x is a maximal planar graph without separating triangles.

The following theorem presents our main result in this section. Theorem 3.45 If G is a maximal planar graph with exactly six separating triangles, then G has a hamiltonian path. Proof. Let G be a maximal planar graph with exactly six separating triangles and let a, b and c be the vertices which form one separating triangle of the graph G. Now we can apply the operation D. The resulting graph is a maximal planar graph with at most 5 separating triangles. By Theorem 3.38, the resulting graph is hamiltonian. Now the vertex, which was added by the operation D, can be removed. Since this is only one vertex, the new graph G has a hamiltonian path. G is a subgraph of G with |V (G)| = |V (G)| and |E(G)| < |E(G)|. Therefore, the graph G has also a hamiltonian path.

With the previous remark and Theorem 3.38 the next corollary follows immediately. Corollary 3.46 Let G be a maximal planar graph with seven separating triangles and let two separating triangles have one edge in common. Then G has a hamiltonian path. Finally, if G is a maximal planar graph with exactly eight separating triangles, then there is a counter-example, which shows that G has no hamiltonian path, see Figure 3.8. Example 3.47 The graph from Figure 3.8 is a maximal planar graph and has no hamiltonian path. The vertices v4, v6, v8, v10, v12, v13, v15, v16 form a maximal independent vertex set. If a hamiltonian path P exists, then these 8 vertices cannot be adjacent. Hence, only two of the remaining 8 vertices can be adjacent to at most one vertex of the maximal independent vertex set. Since the vertices v1 and v2 are adjacent to only one vertex of the maximal independent vertex set, there cannot exist a hamiltonian path.

3.3 Length of a hamiltonian walk

Deﬁnition 3.48 A closed spanning walk of a graph G is a sequence v0e1v1e2...ekvk(= v0), whose terms are alternately vertices and edges, such that the end vertices of ei are vi−1 and vi for each 1 ≤ i ≤ k, and every vertex of G appears in this sequence at least once. The integer k is the length of the closed spanning walk. A hamiltonian walk of G is a closed spanning walk of minimum length. For a connected graph G, h(G) denotes the length of a hamiltonian walk of G. 3.4. TOUGH GRAPHS 41

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Since not every maximal planar graph has a hamiltonian cycle nor a hamiltonian path, we want to determine the length of a hamiltonian walk of a maximal planar graph. A trivial lower bound and a trivial upper bound are known on the length h(G) of a hamiltonian walk of a connected graph G with n vertices:

n ≤ h(G) ≤ 2(n − 1).

These bounds follow immediately from the fact that any hamiltonian walk must pass through every vertex at least once and a closed spanning walk traversing twice every edge of any tree of a connected graph is of length 2(n − 1).

Theorem 3.49 (Asano, Nishizeki and Watanabe [4]) The length h(G) of a hamiltonian walk of any maximal planar graph G with n vertices satisﬁes

≤ 3 (n − 3) if n ≥ 11,  2 h(G)  (3.1) = n otherwise.  3.4 Tough graphs

In this section we consider the question of which maximal planar graphs are non-hamiltonian. Sometimes there is an easily verifiable proof of the fact that a particular graph is non- hamiltonian. We would like to have a theorem that says a graph G is non-hamiltonian if G has property “Q”, where “Q” can be checked in polynomial time. This motivates the definition of 1–toughness. The concept of 1–toughness was introduced by V. Chvátal [21] in 1973. 42 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Deﬁnition 3.50 A graph G is said to be 1–tough if for any non-empty subset S of the vertices of G, κ(G − S) ≤ |S|. Note that every hamiltonian graph is 1–tough. Thus 1–toughness is a necessary condition for a graph to be hamiltonian. Since there exist 1–tough but non-hamiltonian graphs, 1–toughness is not a suﬃcient condition in general.

In 1979 V. Chvátal raised the following question: Is 1–toughness a sufficient condition for a maximal planar graph to be hamiltonian? In 1980 T. Nishizeki [41] answered Chvátal’s question by constructing a 1–tough non-hamiltonian maximal planar graph on 19 vertices. In [25] M. B. Dillencourt found a graph with the same property on 15 vertices. In 1994 M. Tkáˇc [52] found a 1–tough non-hamiltonian maximal planar graph on 13 vertices, see Figure 3.10.

3.5 Construction of maximal planar graphs

This section is concerned with the construction of hamiltonian maximal planar graphs with n vertices and k separating triangles and non-hamiltonian maximal planar graphs with n vertices and k separating triangles. Since maximal planar graphs with more than ﬁve separating triangles are not necessarily hamiltonian, we want to show that there exist at least two maximal planar graphs with the same number of vertices and the same number of separating triangles, such that one graph is hamiltonian and another graph is non-hamiltonian. We start by using Theorem 2.14 to construct a non-hamiltonian MPG-3 graph. Corollary 3.51 Let n ∈ IN with n ≥ 13. Then there exists a non-hamiltonian MPG-3 graph G with n vertices. Proof. We distinguish the following two cases.

Case 1: n is odd. As our basic graph we select an arbitrary maximal planar graph G n−1 with 2 vertices which contains no separating triangles. Since n ≥ 13 this graph G has n+1 at least eight faces. Now we select 2 faces of the graph G. Now we consider the graph G0 obtained from G by inserting a new vertex into each selected face of G then joining the new vertex to every vertex incident with the corresponding face. The number of vertices 0 n+1 0 with degree 3 of G is τ3 = 2 . Then G fulﬁls the assumption 2τ3 > n of Theorem 2.14. Therefore, G0 is non-hamiltonian.

Case 2: n is even. As our basic graph we select an arbitrary maximal planar graph G n−2 with 2 vertices which contains no separating triangles. Since n ≥ 14 this graph G has n+2 at least eight faces. Now we select 2 faces of the graph G. Now we consider the graph G0 obtained from G by inserting a new vertex into each selected face of G then joining the new vertex to every vertex incident with the corresponding face. The number of vertices 0 n+2 0 with degree 3 of G is τ3 = 2 . Then G fulﬁls the assumption 2τ3 > n of Theorem 2.14. Therefore, G0 is non-hamiltonian. 3.5. CONSTRUCTION OF MAXIMAL PLANAR GRAPHS 43

Note that the number of vertices with degree three, τ3, is a lower bound on the number n+1 of separating triangles. If the number of vertices n is odd, we have at least k = 2 n+2 separating triangles. If n is even, we have at least k = 2 separating triangles. Now we consider hamiltonian MPG-3 graphs. Theorem 3.52 Let d ∈ IN. Then there exists a hamiltonian MPG-3 graph G with exactly τ3 = d vertices of degree three.

G1 z0

x0 x y

Figure 3.9: A maximal planar graph G with exactly one vertex of degree 3.

Proof. We construct an MPG-3 graph Gd with exactly d vertices of degree three in- 0 ductively. Therefore, we need a basic graph G1, see Figure 3.9. In G1 the vertices x , y and z0 in Figure 3.9 were identiﬁed with the vertices x, y and z of a copy of the graph 0 0 G1. That is, the face x , y, z is replaced by the graph G1 itself. The resultant graph is called G2. This graph has exactly two vertices of degree three. Now we continue inductively. The resultant graph Gd has exactly d vertices of degree three. Let Bd be the rooted decomposition tree of the given embedding of Gd. Since each vertex of the tree Bd has at most two children, Theorem 3.28 yields that the graph Gd is hamiltonian.

Now we can add the number of separating triangles of Corollary 3.51. If the number n+1 n+2 of vertices n is odd, we have d = 2 separating triangles. If n is even, we have d = 2 separating triangles. But we get a diﬀerent number of vertices. Let n0 be the number of vertices of a graph constructed by Theorem 3.52. Then n + 1 n0 = 7 + 4 · d = 7 + 4 · = 2n + 9 2 if n is odd and n + 2 n0 = 7 + 4 · d = 7 + 4 · = 2n + 11 2 if n is even. In both cases n =6 n0. The next example shows a non-hamiltonian maximal planar graph which was assumed for a long time to be the one with the smallest number of separating triangles. This is not the case, see Figure 3.5. But we need this graph for further constructions. 44 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Example 3.53 Figure 3.10 shows a non-hamiltonian maximal planar graph with 7 separating triangles. The graph G of Figure 3.10 has n = 13 vertices and τ3 = 7 vertices of degree 3 and hence G fulﬁlls the assumption of Theorem 2.14 that 2τ3 > n. Therefore, G

is non-hamiltonian.

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Figure 3.10: A non-hamiltonian maximal planar graph with 7 separating triangles.

In the following we try to adjust k and n. Let n be the number of vertices and k be the number of separating triangles determined. We will show that there are hamiltonian maximal planar graphs with n vertices and k separating triangles and also non-hamiltonian maximal planar graphs with n vertices and k separating triangles. First we will show which values for k are possible.

Lemma 3.54 Let G be the set of all maximal planar graphs with n vertices. Then 0 ≤ k ≤ n − 4 for all G ∈ G.

Proof. Theorem 3.64 of Hakimi and Schmeichel yields the following bounds on the number of cycles of length three: 2n − 4 ≤ C3 ≤ 3n − 8. On the other hand the number of faces is l = 2n − 4. Since a separating triangle does not form the boundary of a face, the number of separating triangles is the diﬀerence between the number of 3–cycles and the number of faces. Hence, we get: 0 ≤ k ≤ n − 4 .

To construct a non-hamiltonian maximal planar graph G0, we must ﬁnd a hamiltonian maximal planar graph G that has a face which has no edge in common with every constructible hamiltonian cycle of G. That is, the face of G must be non-frontier with respect to all hamiltonian cycles of G. 3.5. CONSTRUCTION OF MAXIMAL PLANAR GRAPHS 45

Example 3.55 The graph from Figure 3.11 is a hamiltonian maximal planar graph. C =

v1v4v5v6v8v9v10v12v2v11v3v7v1 is a possible hamiltonian cycle of G

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This provides the following lemma.

Lemma 3.56 Let G be the graph from Figure 3.11. Then the triangle v1v2v3 is a non- frontier face with respect to all hamiltonian cycles of G.

Proof. We will show that neither two nor one of the edges of the triangle v1v2v3 can belong to a hamiltonian cycle of G. Since the graph G is symmetrically developed around the triangle v1, v2, v3, we can without loss of generality select exactly one or exactly two arbitrary edges.

Case 1: The edges v1v2 and v2v3 belong to a hamiltonian cycle C of G. Since the edge v1v3 do not belong to C, one of the edges v7v1 and v7v3 must belong to C. Without loss of generality, the edge v7v1 belongs to C. Now we consider the vertex v4. Since the edges v1v4 and v2v4 would destroy the cycle C, these edges cannot belong to C. Therefore, only the edge v5v4 is incident to the vertex v4. This is a contradiction to the assumption that there exists a hamiltonian cycle C of G which contains the edges v1v2 and v2v3.

Case 2: The edge v1v2 belongs to a hamiltonian cycle C of G and the edges v2v3 and v3v1 do not belong to C. Therefore, one of the edges v4v1 and v4v2 must belong to C. 46 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Without loss of generality, the edge v4v1 belongs to C. Then the edge v4v5 must also belong to C. Now we consider the vertex v7. Since the edge v1v7 would destroy the cycle C, this edge cannot belong to C. Therefore, the edges v8v7 and v3v7 belong to C. In order to merge the vertex v6 into the cycle C, the edges v5v6 and v8v6 must also belong to C. Since v3v11 and v2v11 cannot both belong to the cycle C, the edge v10v11 must belong to the cycle C. Now we consider the vertex v12. The vertex v12 can only be run through by the edges v10v12 and v2v12. So far we have not run through the vertex v9. However, there is no possibility that the cycle C is closed and the vertex v9 will be run through. Thus we get a contradiction and the proof is complete.

Since we need a hamiltonian maximal planar graph G to construct a non-hamiltonian maximal planar graph G0, we must construct a hamiltonian maximal planar graph ﬁrst.

Construction 3.57

Input: n ∈ IN and k ∈ IN with n ≥ k + 6.

Output: A hamiltonian maximal planar graph G with n vertices and k separating triangles.

S1: Let n = k + 5 + r. Select the graph from Figure 3.12 as starting graph G with k + 2 interior vertices.

S2: Consider the triangle abd2. Insert the vertices x1, x2, ..., xr on the edge d1b as follows. The edge d1b is transferred into r+1 edges with vertices d1, x1, x2, ..., xr and b. Connect the vertex c and the vertex d2 with all vertices already inserted.

Proof. Let n ∈ IN and k ∈ IN with n ≥ k + 6. Let n = k + 5 + r. The vertices a, di and b for 1 ≤ i ≤ k+1 form k+1 separating triangles in the starting graph. Since r ≥ 1, at least the vertex xr destroys the separating triangle ad1b. This means we have only k separating triangles. Now we have to show that the constructed graph has a hamiltonian cycle. Begin with the edge adk+2. Then run through the vertices dk+1, ..., d1, x1, x2, ..., xr, b, c, a. Thus we obtain a hamiltonian cycle.

Now we can construct a non-hamiltonian maximal planar graph.

Construction 3.58

Input: n ∈ IN and k ∈ IN with k > 6 and n ≥ k + 6.

Output: A non-hamiltonian maximal planar graph G with n vertices and k separating triangles. 3.5. CONSTRUCTION OF MAXIMAL PLANAR GRAPHS 47

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Figure 3.12: Starting graph G = (V, E).

S1: Let n = k + 5 + r with k = 6 + q. Select the graph from Figure 3.11 as starting graph G.

S2: If q > 1, then use Construction 3.57 to construct a hamiltonian maximal planar graph G0 with (q − 1) + r + 5 vertices and q − 1 separating triangles. 0 Else if q = 1 then choose G as the K4 graph. 0 S3: Replace the triangle v1v2v3 by the graph G .

Proof. Let n ∈ IN and k ∈ IN with k > 6 and n ≥ k + 6. Let n = k + 5 + r with k = 6 + q. If q = 1, then the constructed graph is the graph from Example 3.53. Hence, we have k = 7 separating triangles. If q > 1, then the starting graph G has 6 separating triangles and the constructed hamiltonian maximal planar graph G0 has q − 1 separating triangles. Since the vertices v1v2v3 form another separating triangle, we have exactly k = 6 + q − 1 + 1 = 6 + q separating triangles. Now we have to show that the constructed graph has no hamiltonian cycle. If q = 1, then the constructed graph is the graph from Example 3.53. Hence, we know this graph has no hamiltonian cycle. If q > 1, then the constructed graph is also non-hamiltonian. Since there is no hamiltonian cycle if there is one vertex inside the triangle v1v2v3, there is no hamiltonian cycle, especially if there are more than one vertices inside the triangle v1v2v3.

With the previous constructions we are able to adjust k and n, if n ≥ 13 and k ≥ 7 with n ≥ k + 6. Thus we get the next theorem.

Theorem 3.59 Let n be the number of vertices and k be the number of separating triangles. If n ≥ 13 and k ≥ 7 with n ≥ k + 6, then at least one hamiltonian maximal planar graph with n vertices and k separating triangles exists; and also at least one non- hamiltonian maximal planar graph with n vertices and k separating triangles exists. 48 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Note that the Construction 3.58 works starting from seven separating triangles, whereas Construction 3.57 works also without separating triangles.

Remark. The condition n ≥ k + 6 can be intensiﬁed on n = k + 4 for hamiltonian maximal planar graphs. For the case n = k + 5 there is no construction possible.

Now we consider the case that we have exactly six separating triangles. Note that Figure 3.5 shows a non-hamiltonian maximal planar graph with six separating triangles. This graph provides the basis for our next construction.

v16 v17 v15 v6 v4 v12 v10 v14

v9 v11 v13

v1 v2

Figure 3.13: A non-hamiltonian maximal planar Graph G with n = 17 and k = 6.

Construction 3.60

Input: n ∈ IN and k ∈ IN with k = 6 and n ≥ 17.

Output: A non-hamiltonian maximal planar graph G with n vertices and 6 separating triangles.

S1: Let n = 17 + r. Select the graph from Figure 3.13 as starting graph G.

S2: If r ≥ 2, then use Construction 3.57 to construct a hamiltonian maximal planar graph G0 with r + 4 vertices and no separating triangles. 3.5. CONSTRUCTION OF MAXIMAL PLANAR GRAPHS 49

Else if r = 1 then insert the vertex xr on the edge v7v8 and connect the vertex v4 and the vertex v6 with the new vertex xr. 0 S3: Delete the vertex v17 and replace the triangle v4v10v16 by the graph G .

Proof. Let n ∈ IN and k ∈ IN with k = 6 and n ≥ 17. Let n = 17 + r. If r = 0, then the constructed graph is the graph from Example 3.40. Hence, we have k = 6 separating triangles. If r = 1, then the starting graph G has 6 separating triangles and the inserted vertex xr yields no further separating triangle. If r > 1, then the starting graph G has 6 separating triangles and the inserted graph G0 yields no further separating triangle. Now we have to show that the constructed graph has no hamiltonian cycle. If r = 0, then the constructed graph is the graph from Example 3.40. Hence, we know this graph has no hamiltonian cycle. If r > 1, then the constructed graph is also non-hamiltonian. Since there is no hamiltonian cycle if there is one vertex inside the triangle v4v10v16, there is no hamiltonian cycle, especially if there are more than one vertices inside the triangle v4v10v16. If r = 1, then the vertices v1, v5, v11, v12, v13, v14, v15, v17, xr form a maximal independent vertex set. If a hamiltonian cycle C exists, then these 9 vertices cannot be adjacent. Hence, each of the remaining 9 vertices must be adjacent to at least two vertices of the maximal independent vertex set. Since the vertices v2 and v3 are adjacent to only one vertex of the maximal independent vertex set, there cannot exist a hamiltonian cycle.

With the previous construction and Construction 3.57 we are able to adjust n for k = 6, if n ≥ 17. Thus we get the next theorem.

Theorem 3.61 Let n be the number of vertices and k be the number of separating triangles. If n ≥ 17 and k = 6, then there exists at least one hamiltonian maximal planar graph with n vertices and six separating triangles and also at least one non-hamiltonian maximal planar graph with n vertices and six separating triangles.

The previous remark yields us to the following.

Theorem 3.62 Let n be the number of vertices and k be the number of separating triangles. If n = k + 5, then no maximal planar graph with n vertices and k separating triangles exists.

Proof. Note that each separating triangle has at least one vertex in its interior face and at least one vertex in its exterior face. Since each separating triangle consists of three vertices, we have at most n = 5k vertices. If the separating triangles are not disjoint like Figure 3.12, then we have at least n = k + 4 vertices. To get n = k + 5 vertices we have to add one more vertex. If we add this vertex to an edge, then we have to add two edges to the common neighbors of the two adjacent vertices of the added vertex. If the edge to which we add the new vertex belongs to a separating triangle, then we destroy this separating triangle and get k−1 separating triangles. If the edge to which we add the new vertex does not belong to a separating triangle, then we create a new separating triangle and get k + 1 separating triangles. If we add the new vertex to a face, then we have to 50 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS add three edges to the three vertices of the boundary of the face. Now these three vertices of the boundary of the face form a new separating triangle and we get k + 1 separating triangles.

Analogously to Theorem 2.14 the number of disjoint separating triangles supplies a necessary condition of whether a graph is hamiltonian. Theorem 3.63 Let G = (V, E) be an arbitrary maximal planar graph with k disjoint separating triangles and n ≥ 13 vertices. If 2k > n, then G is non-hamiltonian.b b Proof. Assume that G is hamiltonian. Let C be the hamiltonian cycle of G. Let

D1, ..., Dkb be all disjoint separating triangle of G. Let a, b and c be the vertices of the separating triangle D1. Let the vertices a and b be the neighbors of the interior vertices of

Gin,D1 on the hamiltonian cycle C. If we replace these edges by the edge ab, then we obtain 0 a hamiltonian cycle C in the maximal planar graph G − Gin,D1 . If we delete all disjoint separating triangles of G in this way, we obtain a maximal planar graph G . out,D1,...,Dkb For two disjoint separating triangles we obtain diﬀerent edges. After construction the hamiltonian cycle of G has at least k edges and G consists of at most out,D1,...,Dkb out,D1,...,Dkb n − k vertices. Therefore, k ≤ n − k if and onlyb if 2k ≤ n. This is a contradiction to the assumption.b b b b

3.6 Number of cycles of certain length

In this section we consider the following question. What is the number of cycles of certain length that a maximal planar graph G on n vertices could have, in terms of n?

3.6.1 Number of short cycles

Let G be a maximal planar graph on n vertices, and let Ci(G) for 3 ≤ i ≤ n denote the number of cycles of length i in G. We ﬁrst present tight bounds for C3(G) and C4(G) in terms of n. In 1979 S. L. Hakimi and E. F. Schmeichel proved the following. Theorem 3.64 (Hakimi and Schmeichel [31]) Let G be a maximal planar graph with n ≥ 6 vertices. Then 2n − 4 ≤ C3(G) ≤ 3n − 8. The lower bound is attained if and only if G is 4–connected, and the upper bound is attained if and only if G is obtained from K3 by recursively placing a vertex of degree 3 inside a face, and joining this new vertex to the three vertices incident to that face. Note that no n-vertex maximal planar graph G with C3(G) ≤ 3n − 9 exists. For 4–cycles S. L. Hakimi and E. F. Schmeichel achieved the following result. Theorem 3.65 (Hakimi and Schmeichel [31]) Let G be a maximal planar graph with n ≥ 5 vertices. Then 1 3n − 6 ≤ C (G) ≤ (n2 + 3n − 22). 4 2 3.6. NUMBER OF CYCLES OF CERTAIN LENGTH 51

The lower bound is attained if and only if n = 5 or G is 5–connected. The upper bound is attained if and only if G is the graph in Figure 3.12. If the graph is additionally 4–connected, the following upper bound holds.

Lemma 3.66 (Hakimi and Schmeichel [31]) Let G be a 4–connected maximal planar graph with n ≥ 7 vertices. Then 1 C (G) ≤ (n2 − n − 2). 4 2 Note that equality in Lemma 3.66 holds if and only if G is the graph of Figure 3.15.

3.6.2 Number of longer cycles Now we continue with the number of longer cycles. We interpret longer cycles as cycles starting from the length ﬁve. It was also Hakimi and Schmeichel who established a ﬁrst lower bound for maximal planar graph on at least 8 vertices.

Lemma 3.67 (Hakimi and Schmeichel [31]) Let G be a maximal planar graph with n ≥ 8 vertices. Then C5(G) ≥ 6n.

There exists a maximal planar graph G with C5(G) = 6n for every n ≥ 14. An upper bound was found for maximal planar graphs on at least 6 vertices.

Lemma 3.68 (Hakimi and Schmeichel [31]) Let G be a maximal planar graph with n ≥ 6 vertices. Then 2 C5(G) ≤ 5n − 26n.

Now we consider an upper bound for Ck(G) for an arbitrary k.

Theorem 3.69 (Hakimi and Schmeichel [31]) Let G be a maximal planar graph with n b k c vertices. Then Ck(G) ∈ O(n 2 ), for k = 3, 4, ..., n.

3.6.3 Number of hamiltonian cycles As there are maximal planar graphs that are non-hamiltonian, there are maximal planar graphs with Cn(G) = 0. Therefore, let G be a hamiltonian maximal planar graph on n vertices. Then the following question occurs. What is the minimum number of hamiltonian cycles that G could have in terms of n? First we will consider 3–connected maximal planar graphs.

Theorem 3.70 (Hakimi, Schmeichel and Thomassen [32]) Let n ≥ 12. Form a maximal planar graph G with n vertices as follows: Begin with the graph G2 of Figure 3.14 (with i = n − 11), and in the interior of the triangle T = v1v2v3 place the graph G1 of Figure 3.14 so that G1 and G2 have precisely the triangle T in common. Then G has exactly four hamiltonian cycles.

52 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS ¢

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We observe that maximal planar graphs constructed in Theorem 3.70 all have connectivity three. It seems natural, therefore, to inquire about the minimum number of hamiltonian cycles in 4–connected maximal planar graphs on n vertices. It is known that every such graph has at least one hamiltonian cycle.

Theorem 3.71 (Hakimi, Schmeichel and Thomassen [32]) Let G be a 4–connected maximal planar graph with n vertices. Then G contains at least n hamiltonian cycles. log2 n

The lower bound given in Theorem 3.71 occurs far from tight. In fact we oﬀer the following.

Example 3.72 Let G be the 4–connected maximal planar graph with n vertices of Figure 3.15. Then G contains at least 2(n − 2)(n − 4) hamiltonian cycles. Assume that G is the 4–connected maximal planar graph of Figure 3.15 with n = 9 vertices. Then G contains at least 70 hamiltonian cycles. Theorem 3.71 yields G contains at least three hamiltonian cycles.

Now we prove an improved lower bound on the number of hamiltonian cycles.

Theorem 3.73 Let G be a 4–connected maximal planar graph with n ≥ 7 vertices. Then G contains at least 4n − 8 hamiltonian cycles.

Proof. Let G = (V, E) with |V | = n and |E| = m be a 4–connected maximal planar graph and let l(G) be the number of faces of G. Then Theorem 1.1 of Euler provides l(G) = m − n + 2. Moreover, 3l(G) = 2m, since each edge is located on the boundary of two faces. Then with Theorem 1.1 of Euler we obtain l = 2n − 4. Since G is 4–connected, G contains no separating triangles. By Theorem 3.36 there exists a hamiltonian cycle C 3.6. NUMBER OF CYCLES OF CERTAIN LENGTH 53

of G such that k2F = 4. This means each face of G is incident to at least one edge of the hamiltonian cycle C. Since the embedding of a 4–connected maximal planar graph is unique, we embed G such that three other vertices form the exterior face. If we use the same geometric structure as C, we get a new hamiltonian cycle C 0. If we do this for each face of G, we get 2n − 4 diﬀerent hamiltonian cycles. Since a hamiltonian cycle has two diﬀerent orientations, we get 2(2n − 4) = 4n − 8 hamiltonian cycles.

The next example will show that the lower bound given in the previous theorem appears not so far from tight.

Example 3.74 Consider Example 3.72 again. Assume that G is a 4–connected maximal planar graph of Figure 3.15 with n = 9 vertices. Theorem 3.73 yields G contains at least 28 hamiltonian cycles.

Figure 3.15: A maximal planar graph.

The previous results refer to 4–connected maximal planar graph. Now we consider 3–connected maximal planar graph with at most 5 separating triangles, since we know these graphs are hamiltonian. Let G = (V, E) be a 3–connected maximal planar graph and let D1, D2, ..., Dk for 1 ≤ k ≤ 5 be the separating triangles of G. Then we deﬁne ns as the number of all vertices inside of all separating triangles.

ns = |{v ∈ V (G) | v ∈ V (G) \ V (Gout,D1,D2,...,Dk )}| for 1 ≤ k ≤ 5.

Theorem 3.75 Let G be a maximal planar graph with at most 5 separating triangles and with n − ns ≥ 7 vertices. Then G contains at least 4(n − ns) − 8 hamiltonian cycles.

Proof. Let G = (V, E) with |V | = n and |E| = m be a maximal planar graph with at most 5 separating triangles and let D1, D2, ..., Dk for 1 ≤ k ≤ 5 be the separating triangles of G. Now consider the graph Gout,D1,D2,...,Dk with n − ns vertices and without separating triangles. By Theorem 3.36 there exists a hamiltonian cycle C of Gout,D1,D2,...,Dk such 54 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

that k2F = 4. This means each face of Gout,D1,D2,...,Dk is incident to at least one edge of the hamiltonian cycle C. Since the embedding of a 4–connected maximal planar graph is unique, we embed Gout,D1,D2,...,Dk such that three other vertices form the exterior face. If we use the same geometric structure as C, we get a new hamiltonian cycle C 0. If we do this for each face of Gout,D1,D2,...,Dk , we get 2(n−ns)−4 different hamiltonian cycles. Since a hamiltonian cycle has two different orientations, we get 2(2(n − ns) − 4) = 4(n − ns) − 8 hamiltonian cycles. Now we have to insert the ns deleted vertices. We distinguish the following five cases.

Case 1: The graph H = Gin,D1 has four separating triangles. By Theorem 3.37 H is hamiltonian. Let the vertices a, b and c be the vertices of the separating triangle D1. If 00 Hout,D2,...,D5 has n < 7 vertices, then there exists a hamiltonian cycle C of Hout,D2,...,D5 such that the hamiltonian cycle passes through at least one of the edges ab, bc and ca. If

Hout,D2,...,D5 has n ≥ 7 vertices, then by Theorem 3.36 there exists a hamiltonian cycle 00 C of Hout,D2,...,D5 such that k2F = 4 and at least two 2-frontier faces have one edge in common. Now we have to prove that the hamiltonian cycle C 00 passes through at least one edge of the edges ab, bc and ca. We insert 4 new vertices each into the facial cycles

D2, D3, D4 and D5 of Hout,D2,...,D5 then joining the new vertex to every vertex incident with 0 the faces of D2, D3, D4 and D5 respectively. The new graph H is a hamiltonian maximal planar graph with 4 separating triangles. Now we can extend the hamiltonian cycle C 00 of 000 0 000 000 Hout,D2,...,D5 to a hamiltonian cycle C of H . Thus we get k2F (C ) = 7 and kN (C ) = 3 with respect to the hamiltonian cycle C000. Then we can embed the graph H 0 on a sur- 000 face such that D1 is not the non-frontier face. Therefore, the hamiltonian cycle C passes through at least one edge of the edges ab, bc and ca. At last we have to replace the vertices in the facial cycles D2, D3, D4 and D5 by the graphs Gin,D2 − D2, Gin,D3 − D3, Gin,D4 − D4 and Gin,D5 − D5 respectively. Since Gin,Di with 2 ≤ i ≤ 5 are maximal planar graph without separating triangles, the graphs Gin,Di with 2 ≤ i ≤ 5 are hamiltonian for any two boundary edges of Gin,Di . Therefore, the graphs Gin,Di − Di with 2 ≤ i ≤ 5 have a hamiltonian path in Gin,Di −Di. Now we replace the two edges which join the new vertices 000 to the hamiltonian cycle C by the hamiltonian paths in Gin,Di − Di with 2 ≤ i ≤ 5. Thus we get a hamiltonian cycle Civ which passes through at least one of the edges ab, bc iv and ca. C is a hamiltonian cycle of Gin,D1 . Without loss of generality, Gin,D1 has a hamiltonian cycle Cin = abcPin(c, a)a, where Pin(c, a) is a hamiltonian path of Gin,D − b between c and a. Without loss of generality, Civ passes through the edge ca. Then we iv replace the edge ca in the hamiltonian cycle C of Gout,D1,D2,...,D5 by the hamiltonian path

Pin(c, a) between c and a of Gin,D1 − b. Then the new cycle is a hamiltonian cycle of G.

Case 2: The graph H = Gin,D1 has three separating triangles D2, D3, D4. By Theorem 3.18 H is hamiltonian. Then analogically to Case 1 the graph G has a hamiltonian cycle.

Case 3: The graph H = Gin,D1 has two separating triangles D2, D3. By Theorem 3.17 H is hamiltonian. Then analogically to Case 1 the graph G has a hamiltonian cycle.

Case 4: The graph H = Gin,D1 has one separating triangle D2. By Theorem 3.16 H is hamiltonian. Then analogically to Case 1 the graph G has a hamiltonian cycle. 3.7. PANCYCLIC MAXIMAL PLANAR GRAPH 55

Case 5: The graph H = Gin,D1 has no separating triangles. By Theorem 3.3 H is hamiltonian. Then analogically to Case 1 the graph G has a hamiltonian cycle. All the other mixed cases arise from the remaining cases. The number of hamiltonian cycles does not change, because we have only one edge replaced with a hamiltonian path in all cases.

Note that the previous result is not true for all maximal planar graphs with at most ﬁve separating triangles. As the condition n − ns ≥ 7 is very incising in the following sense. The set {v ∈ V (G) | v ∈ V (G) \ V (Gout,D1,D2,...,Dk )} can contain a lot of vertices, in the worst case ns = n − 4. The worst case occurs if the separating triangles are not disjoint. Moreover, if there are at least three disjoint separating triangles, Theorem 3.75 always holds.

As there are maximal planar graphs with more than ﬁve separating triangles that are non-hamiltonian, there are maximal planar graphs with Cn(G) = 0. Therefore, let G be a hamiltonian maximal planar graph on n vertices with k separating triangles for 6 ≤ k ≤ 8. Then the following question remains open. What is the minimum number of hamiltonian cycles that G could have in terms of n?

3.7 Pancyclic maximal planar graph

In this section we move away from the property that a graph contains one cycle of length only n and turn towards a problem determining when a graph G contains a cycle of each possible length l with 3 ≤ l ≤ n.

Deﬁnition 3.76 A graph G = (V, E) on n vertices is called pancyclic if it contains a cycle of each possible length l with 3 ≤ l ≤ n.

First we will prove that the more general class of hamiltonian planar triangulations is pancyclic.

Theorem 3.77 Let G = (V, E) be a hamiltonian planar triangulation. Then G is pancyclic.

Proof. For the proof we need the following two claims.

Claim 1. A planar triangulation T , whose vertices are all on XT , has at least two vertices of degree 2.

Proof of Claim 1. Consider a chord which separates the graph T into two parts. Choose one of those subgraphs and call this graph T1. Now consider inductively also a chord of T1. Continuing this approach, we get a graph which is a triangle. Since one vertex has the same degree as in the graph T , this vertex has degree 2. We can ﬁnd the second vertex with degree 2 if we consider the other subgraph of the ﬁrst separation. 56 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Claim 2. A planar triangulation T whose vertices are all on XT is pancyclic.

Proof of Claim 2. For the graph T = (V, E) with |V | = |XT | = n the exterior cycle XT is also a hamiltonian cycle. It follows from Claim 1 that there are at least two vertices of degree 2. Let v be one of these vertices. Let v1 and v2 be the vertices which are both adjacent to v. Since the boundary of every face of T is a triangle, except possibly the exterior face, the vertices v1 and v2 are adjacent. We replace the edges vv1 and vv2 0 in the hamiltonian cycle C with the edge v1v2, then we get a cycle C of length n − 1. Consider the graph T − v. This graph is also a planar triangulation with a hamiltonian cycle C0. Continuing this approach, we get cycles of length l with 3 ≤ l ≤ n. Therefore, T is pancyclic.

Let G be a hamiltonian planar triangulation. Let C be the hamiltonian cycle of G. We generate a new graph G0 by deleting all edges which are outside of the hamiltonian 0 cycle C. We get a planar triangulation G , whose vertices are all on XG0 . It follow from

Claim 2 that G0 is pancyclic. Then G is also pancyclic.

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This theorem yields that the class of hamiltonian maximal planar graphs is also pancyclic.

Corollary 3.78 Let G = (V, E) be a hamiltonian maximal planar graph. Then G is pancyclic.

As not all maximal planar graphs are hamiltonian, we seek a stronger result. In 1979 S. L. Hakimi and E. F. Schmeichel [31] proved the following.

Theorem 3.79 (Hakimi and Schmeichel [31]) Let G = (V, E) be a maximal planar graph. Let r be the length of the longest cycle in G. Then G contains a cycle of each possible length l, 3 ≤ l ≤ r.

Analogously to Deﬁnition 3.76 we consider the special case of pancyclic modulo k. 3.8. DELETION OF VERTICES 57

Deﬁnition 3.80 For any integer k, we deﬁne a graph G = (V, E) to be pancyclic modulo k if it contains a cycle of every length modulo k.

In 1991 N. Dean [23] showed the next result.

Theorem 3.81 (Dean [23]) Let G = (V, E) be a 3–connected planar graph (except K4) with minimum degree at least k. Then G is pancyclic modulo k.

This result yields the following corollary.

Corollary 3.82 i) Each MPG-4 graph is pancyclic modulo 4.

ii) Each MPG-5 graph is pancyclic modulo 5.

3.8 Deletion of vertices

M. D. Plummer [42] has conjectured that any graph obtained from a 4–connected planar graph by deleting one vertex has a hamiltonian cycle. This conjecture follows from Tutte’s Theorem as observed by D. A. Nelson [40]. In fact it also follows from Lemma 3.7 of C. Thomassen [51] that the deletion of any vertex from a 4–connected planar graph results in a hamiltonian graph.

Theorem 3.83 (Thomassen [51]) Let G be a 4–connected planar graph and let x be a vertex of G. Then G − x is hamiltonian.

D. A. Nelson [40] extended Thomassen’s Theorem 3.83.

Theorem 3.84 (Nelson [40]) Let G be a 4–connected planar graph and let x and y be any two vertices of G that lie on the same face boundary. Then G − x − y is hamiltonian.

M.D. Plummer [42] also conjectured that any graph obtained from a 4–connected planar graph by deleting two vertices has a hamiltonian cycle. This conjecture was proved by R. Thomas and X. Yu [50].

Theorem 3.85 (Thomas and Yu [50]) Let G be a 4–connected planar graph and let x and y be two vertices of G. Then G − x − y is hamiltonian.

Note that deleting three vertices from a 4–connected planar graph may result in a graph which is not 2–connected and hence has no hamiltonian cycle. However, D. P. Sanders [45] obtained a stronger version of Thomas and Yu’s result. This follows from Theorem 3.11 of D. P. Sanders.

Theorem 3.86 (Sanders [45]) Let G be a 4–connected planar graph and let x, y and z be the vertices of a triangle in G. Then G − x − y − z is hamiltonian.

Now the following question occurs; what happens if we delete one vertex from a 3– connected maximal planar graph with separating triangles? 58 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Theorem 3.87 (Birchel and Helden [10]) Let G be a maximal planar graph with exactly one separating triangle and let x be a vertex of G. Then G − x is hamiltonian.

Proof. Let a, b and c be the vertices of the separating triangle D.

Case 1: Let x 6∈ D.

Subcase 1.1: Let x ∈ Gout,D. Consider Gin,D, a, b and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gin,D is hamiltonian for any two boundary edges. Without loss of generality, Gin,D has a hamiltonian cycle Cin = abcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − b between c and a. Next consider the graph Gout,D − x. Gout,D − x is a planar triangulation without separating triangles. Since G is 3–connected, Gout,D − x can only be 2–connected if the vertex x is adjacent to a vertex of degree three. Each vertex of degree three represents a simple separating triangle. Since there is only one separating triangle D and x 6∈ D, Gout,D − x is 3–connected. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. It follows from Corollary 4.23 that there exists a hamiltonian cycle Cout of Gout,D −x such that the edge ca is contained in E(Cout). Now we replace the edge ca by the path Pin(c, a) in the graph G−x. Then G−x has a hamiltonian cycle and G−x is hamiltonian.

Subcase 1.2: Let x ∈ Gin,D. Consider the graph Gout,D. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Lemma 1.10. Without loss of generality, Gout,D has a hamiltonian cycle Cout = abcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D − b between c and a. Next consider the graph Gin,D − x. Gin,D − x is a planar triangulation without separating triangles. It follows from Corollary 4.23 that there exists a hamiltonian cycle Cin of Gin,D − x such that the edge ca is contained in E(Cin). Now we replace the edge ca by the path Pout(c, a) in the graph G − x. Then G − x has a hamiltonian cycle and G − x is hamiltonian.

Case 2: Let x ∈ D.

Subcase 2.1: The vertex x is not adjacent to a vertex of degree three. The graph G − x is a 3–connected planar triangulation without separating triangles. It follows from Corollary 4.23 that G − x is hamiltonian.

Subcase 2.2: The vertex x is adjacent to a vertex of degree three. The graph G−x is a 2–connected planar triangulation without separating triangles. Without loss of generality, let x = b. Consider Gin,D, a, x and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gin,D is hamiltonian for any two boundary edges. Without loss of generality, Gin,D has a hamiltonian cycle Cin = axcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − x between c and a. Consider the graph Gout,D. Note that a planar graph 3.8. DELETION OF VERTICES 59 can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Lemma 1.10. Without loss of generality, Gout,D has a hamiltonian cycle Cout = axcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D −x between c and a. Now we connect the two paths Pin(c, a) and Pout(c, a) to one cycle. This is a hamiltonian cycle in the graph G − x and the proof is complete

Now we can extend Theorem 3.87 to exactly two separating triangles.

Theorem 3.88 (Birchel and Helden [10]) Let G be a maximal planar graph with exactly two separating triangles and let x be a vertex of G. Then G − x is hamiltonian.

Proof. Let D and T be the two separating triangles. Let a, b and c be the vertices of the separating triangle D.

Case 1: x does not belong to any separating triangle.

Subcase 1.1: T is inside the graph Gout,D.

Subcase 1.1.1: Let x ∈ Gout,D. Consider Gin,D. a, b and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gin,D is hamiltonian for any two boundary edges. Without loss of generality, Gin,D has a hamiltonian cycle Cin = abcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − b between c and a. Next consider the graph Gout,D. The graph Gout,D is maximal planar with one separating triangle T . Let e, f and g be the vertices of the separating triangle T . The graph Gout,D,T is maximal planar without separating triangles. Let u, v and x be the vertices of a triangle F of the graph Gout,D,T . It follows from Corollary 3.22 that there exists a hamiltonian cycle Cout of Gout,D,T and edges ge ∈ E(T ) and ca ∈ E(D) such that ux, xv, ge, ca are distinct and contained in E(Cout). Consider the graph Gin,T . Since Gin,T is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gin,T is hamiltonian for any two boundary edges. Without loss of generality, 0 0 0 Gin,T has a hamiltonian cycle Cin = efgPin(g, e)e where Pin(g, e) is a hamiltonian path of Gin,T − f between g and e. Consider the triangle F . Since the hamiltonian cycle Cout of Gout,D,T contains the edges ux and xv, we can delete the vertex x. Since u and v are 0 adjacent, we get a hamiltonian cycle Cout of Gout,D,T − x which contains the edges ca and ge. Now we replace the edge ca by the path Pin(c, a) in the graph G − x and we replace 0 the edge ge by the path Pin(g, e) in the graph G − x. Then G − x has a hamiltonian cycle and G − x is hamiltonian.

Subcase 1.1.2: Let x ∈ Gin,D. Consider the graph Gout,D. Gout,D is a maximal planar graph with one separating triangle. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Lemma 1.10. It follows from Theorem 3.17 that Gout,D is hamiltonian for any two boundary edges. With- out loss of generality, Gout,D has a hamiltonian cycle Cout = abcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D − b between c and a. Next consider the graph Gin,D − x. 60 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Gin,D − x is a planar triangulation without separating triangles. Since G is 3–connected, Gin,D − x can only be 2–connected if the vertex x is adjacent to a vertex of degree three. Each vertex of degree three represents a simple separating triangle. Since there is only one separating triangle D and x 6∈ D, Gin,D − x is 3–connected. a, b and c are the vertices of the triangle D. It follows from Corollary 4.23 that there exists a hamiltonian cycle Cin of Gin,D − x such that the edge ca is contained in E(Cin). Now we replace the edge ca by the path Pout(c, a) in the graph G − x. Then G − x has a hamiltonian cycle and G − x is hamiltonian.

Subcase 1.2: T is inside the graph Gin,D.

Subcase 1.2.1: Let x ∈ Gout,D. Consider Gin,D, a, b and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph with one separating triangle, it follows from Theorem 3.17 that Gin,D is hamiltonian for any two boundary edges. Without loss of generality, Gin,D has a hamiltonian cycle Cin = abcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − b between c and a. Next consider the graph Gout,D. Gout,D − x is a planar triangulation without separating triangles. Since G is 3–connected, Gout,D − x can only be 2–connected if the vertex x is adjacent to a vertex of degree three. Each vertex of degree three represents a simple separating triangle. Since there is only one separating triangle D and x 6∈ D, Gout,D − x is 3–connected. a, b and c are the vertices of the triangle D. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Lemma 1.10. It follows from Corollary 4.23 that there exists a hamiltonian cycle Cout of Gout,D − x such that the edge ca is contained in E(Cout). Now we replace the edge ca by the path Pin(c, a) in the graph G − x. Then G − x has a hamiltonian cycle and G − x is hamiltonian.

Subcase 1.2.2: Let x ∈ Gin,D. Consider Gout,D. Since Gout,D is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gout,D is hamiltonian for any two boundary edges. Without loss of generality, Gout,D has a hamiltonian cycle Cout = abcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D −b between c and a. Next consider the graph Gin,D. Note that Gin,D is a maximal planar graph with one separating triangle T . Let e, f and g be the vertices of the separating triangle T . Deﬁne H = Gin,D and Gin,D,T = Hout,T . Then the graph Gin,D,T is a maximal planar graph without separating triangles. Let F = {u, v, x} be a triangle of the graph Gin,D,T . It follows from Corollary 3.22 that there exists a hamiltonian cycle Cin of Gin,D,T and edges ge ∈ E(T ) and ca ∈ E(D) such that ux, xv, ge, ca are distinct and contained in E(Cin). Consider the graph Gin,T . Since Gin,T is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gin,T is hamiltonian for any two boundary edges. Without loss of generality, Gin,T has a hamiltonian cycle 0 0 0 Cin = efgPin(g, e)e where Pin(g, e) is a hamiltonian path of Gin,T − f between g and e. Consider the triangle F . Since the hamiltonian cycle Cin of Gin,D,T contains the edges ux and xv, we can delete the vertex x. Since u and v are adjacent, we get a hamilto- 0 nian cycle Cin of Gin,D,T − x which contains the edges ca and ge. Now we replace the 3.8. DELETION OF VERTICES 61

edge ca by the path Pout(c, a) in the graph G − x and replace the edge ge by the path 0 Pin(g, e) in the graph G−x. Then G−x has a hamiltonian cycle and G−x is hamiltonian.

Case 2: Without loss of generality, x ∈ D.

Subcase 2.1: T is inside the graph Gout,D. Without loss of generality, let x = b. Consider Gin,D, a, x and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph without separating triangles, it follows from Theorem 3.16 of Chen that Gin,D is hamiltonian for any two boundary edges. Without loss of generality, Gin,D has a hamiltonian cycle Cin = axcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − x between c and a. Consider the graph Gout,D. Gout,D is a maximal planar graph with one separating triangle. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. It follows from Theorem 3.17 that Gout,D is hamiltonian for any two boundary edges. Without loss of generality, Gout,D has a hamiltonian cycle Cout = axcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D − x between c and a. Now we connect the two paths Pin(c, a) and Pout(c, a) to one cycle. This is a hamiltonian cycle in the graph G − x and G − x is hamiltonian.

Subcase 2.2: T is inside the graph Gin,D. Without loss of generality, let x = b. Con- sider Gin,D, a, x and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph with one separating triangle, it follows from Theorem 3.17 that Gin,D is hamiltonian for any two boundary edges. Without loss of generality, Gin,D has a hamiltonian cycle Cin = axcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − x between c and a. Consider the graph Gout,D. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. Without loss of generality, Gout,D has a hamiltonian cycle Cout = axcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D −x between c and a. Now we connect the two paths Pin(c, a) and Pout(c, a) to one cycle. This is a hamiltonian cycle in the graph G − x and the proof is complete.

We can extend Theorem 3.88 to exactly three separating triangles, if we assume that the deleted vertex x belongs to one of the separating triangles. The problem of whether G − x is hamiltonian if the deleted vertex x belongs to the interior or exterior of one separating triangle is still open.

Theorem 3.89 (Birchel and Helden [10]) Let G be a maximal planar graph with exactly three separating triangles and let x be a vertex of one of these separating triangles of G. Then G − x is hamiltonian.

Proof. Let D be one separating triangle and let a, b and c be the vertices of the separating triangle D. Without loss of generality, x ∈ D and without loss of generality, let x = b. Consider Gin,D. a, x and c are the vertices of the exterior cycle of Gin,D. Since Gin,D is a maximal planar graph with at most two separating triangles, it follows from Theorem 3.27 that Gin,D is hamiltonian for any two boundary edges. Without loss of 62 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

generality, Gin,D has a hamiltonian cycle Cin = axcPin(c, a)a where Pin(c, a) is a hamiltonian path of Gin,D − x between c and a. Next consider the graph Gout,D. Gout,D is a maximal planar graph with at most two separating triangles. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Lemma 1.10. It follows from Theorem 3.27 that Gout,D is hamiltonian for any two boundary edges. Without loss of generality, Gout,D has a hamiltonian cycle Cout = axcPout(c, a)a where Pout(c, a) is a hamiltonian path of Gout,D −x between c and a. Now we connect the two paths Pin(c, a) and Pout(c, a)to one cycle. This is a hamiltonian cycle in the graph G − x and G − x is hamiltonian.

The next example shows that the deletion of one vertex from a maximal planar graph with four separating triangles may result in a graph which has no hamiltonian cycle.

Example 3.90 (Birchel and Helden [10]) The left side of Figure 3.17 shows a hamiltonian maximal planar graph G with four separating triangles. The right side of Figure 3.17 shows a non-hamiltonian planar triangulation G − x with one separating triangle after the deletion of the vertex x. Since the adjacent vertices a, b and c have degree two, these vertices imply a cycle. Thus it is not possible to form a hamiltonian cycle in the graph

G − x. Note that the vertex x belongs to three separating triangles.

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Figure 3.17: Hamiltonian maximal planar graph G and non-hamiltonian graph G − x.

Now the following question occurs; what happens if we delete two vertices from a 3–connected maximal planar graph with separating triangles?

Theorem 3.91 (Birchel and Helden [10]) Let G = (V, E) be a hamiltonian maximal planar graph with at least one separating triangle and let x and y be two vertices of one of these separating triangles of G. Then G − x − y is non-hamiltonian.

Proof. Let x, y and z be the vertices of the separating triangle T . Then there is a subgraph Tin = Gin,T − V (T ), see Figure 3.18. After the deletion of x and y there is only 3.8. DELETION OF VERTICES 63

the possibility by z on a hamiltonian cycle to get into the subgraph Tin. After running through the subgraph we must leave it again. However, this is possible only by the vertex z. This leads to a contradiction.

x y x y

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Figure 3.18: Deletion of an edge of a separating triangle.

However, if we assume that exactly one vertex belongs to the separating triangle, we can prove the following.

Theorem 3.92 (Birchel and Helden [10]) Let G = (V, E) be a maximal planar graph with exactly one separating triangle. If exactly one vertex of two vertices x and y belongs to the separating triangle, then G − x − y is hamiltonian.

Proof. Let u, v and x be the vertices of the separating triangle T .

Case 1: The vertex y is inside the graph Gin,T . Consider the graph Gin,T −y. Gin,T −y is a planar triangulation without separating triangles. Since G is 3–connected, Gin,T − y can only be 2–connected if the vertex y is adjacent to a vertex of degree three. Each vertex of degree three represents a simple separating triangle. Since there is only one separating triangle T and y does not belong to T , Gin,T − y is 3–connected. u, v and x are the vertices of the triangle T . It follows from Corollary 4.23 that there exists a hamiltonian cycle Cin of Gin,T − y such that the edge uv is contained in E(Cin). Next consider the graph Gout,T . Gout,T is a maximal planar graph without separating triangles. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. It follows from Theorem 3.16 that Gout,T is hamiltonian for any two boundary edges. Without loss of generality, Gout,T has a hamiltonian cycle Cout = uxvPout(v, u)u where Pout(v, u) is a hamiltonian path of Gout,T −x between v and u. Now we replace the edge uv by the path Pout(v, u) in the graph G−x−y. This is a hamiltonian cycle in the graph G−x−y and G−x−y is hamiltonian.

Case 2: The vertex y is inside the graph Gout,T . Consider the graph Gout,T − y. Gout,T − y is a planar triangulation without separating triangles. Since G is 3–connected, Gout,T − y can only be 2–connected if the vertex y is adjacent to a vertex of degree three. 64 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

Each vertex of degree three represents a simple separating triangle. Since there is only one separating triangle T and y does not belong to T , Gout,T − y is 3–connected. u, v and x are the vertices of the triangle T . It follows from Corollary 4.23 that there exists a hamiltonian cycle Cout of Gout,T − y such that the edge uv is contained in E(Cout). Next consider the graph Gin,T . Gin,T is a maximal planar graph without separating triangles. Note that a planar graph can always be embedded in a surface such that a given face of the graph becomes the exterior face, see Chiba and Nishizeki [19]. It follows from Theorem 3.16 that Gin,T is hamiltonian for any two boundary edges. Without loss of generality, Gin,T has a hamiltonian cycle Cin = uxvPin(v, u)u where Pin(v, u) is a hamiltonian path of Gin,T −x between v and u. Now we replace the edge uv by the path Pin(v, u) in the graph G−x−y. This is a hamiltonian cycle in the graph G−x−y and G−x−y is hamiltonian.

The problem of whether G − x − y is hamiltonian, if the deleted vertices x and y both do not belong to the separating triangle is still open. Theorem 3.92 provides the next immediate corollary, since the vertex which belongs to the separating triangle destroys this separating triangle and we get a planar triangulation without separating triangles.

Corollary 3.93 (Birchel and Helden [10]) Let G be a 3–connected planar triangulation without separating triangles and let x be a vertex of G. Then G − x is hamiltonian.

If for a maximal planar graph G with exactly one separating triangle and for two non-adjacent vertices x and y which do not belong to the separating triangle, the graph G − x − y is hamiltonian, then for a 3–connected planar triangulation G with exactly one separating triangle and for a vertex x, the graph G − x is hamiltonian.

Now we consider two separating triangles. The next example shows that the deletion of two vertices from a maximal planar graph with exactly two separating triangles may result in a graph which has no hamiltonian cycle.

Example 3.94 (Birchel and Helden [10]) The left side of Figure 3.19 shows a hamiltonian maximal planar graph G with two separating triangles. The right side of Figure 3.19 shows a non-hamiltonian planar triangulation G − u − v without separating triangles after the deletion of the non-adjacent vertices u and v. Note that both vertices u and v belong to a separating triangle.

Anyway the following result can be proved.

Theorem 3.95 (Birchel and Helden [10]) Let G be a hamiltonian maximal planar graph with at most two separating triangles T1 and T2 and let x and y be two adjacent vertices of G that do not lie on any separating triangle. If the graph G − x − y is 3–connected, then G − x − y is hamiltonian.

Proof. Since x and y are adjacent, we get a planar triangulation after the deletion of both vertices. If x and y lie on a separating 4-cycle, then G − x − y is only 2–connected. Note that each vertex cut of a maximal planar graph is a cycle. If we assume that there is a chordal 4-cut, a cycle arises, because both edges which lie on the exterior face are

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Figure 3.19: Deletion of two vertices of a maximal planar graph with two separating triangles. incident anyway, because the exterior face is bounded by a 3-cycle. Since G − x − y is 3–connected, G − x − y is hamiltonian by Theorem 4.27.

Now we consider three separating triangles. The next example shows that the deletion of two vertices from a maximal planar graph with exactly three separating triangles may result in a graph which has no hamiltonian cycle.

Example 3.96 (Birchel and Helden [10]) Figure 3.20 shows a hamiltonian maximal planar graph G with three separating triangles. The left side of Figure 3.21 shows a non- hamiltonian planar triangulation G − u − v with one separating triangle after the deletion of the adjacent vertices u and v. The right side of Figure 3.21 shows a non-hamiltonian planar triangulation G − u − w without separating triangles after the deletion of the non- adjacent vertices u and w. Note that all three vertices u, v and w belong to a separating triangle.

An additional condition which we could demand would be that we have only simple separating triangles Ti in the graph. This means Gin,Ti would be the K4, or in other words only vertices of degree three form separating triangles. Now we determine one of these vertices of degree three as one of the vertices to be deleted. Then we can consider that the deletion of a vertex of degree three does not transform a maximal planar graph into a planar triangulation. The consideration for the second vertex, which we would like to delete, can be led back in the case of deletion of only one vertex of a maximal planar graph with i − 1 separating triangles.

3.9 Claw-free maximal planar graph

Deﬁnition 3.97 A graph G is claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph K1,3.

66 CHAPTER 3. HAMILTONICITY OF MAXIMAL PLANAR GRAPHS

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